Cepheids have been the cornerstone of the extragalactic distance scale for a century. With high-quality data, these luminous supergiants exhibit a small dispersion in their Leavitt (period–luminosity) relation, particularly at longer wavelengths, and few methods rival the precision possible with Cepheid distances. In these proceedings, we present an overview of major observational programs pertaining to the Cepheid extragalactic distance scale, its progress and remaining challenges. In addition, we present preliminary new results on Cepheids from the James Webb Space Telescope (JWST). The launch of JWST has opened a new chapter in the measurement of extragalactic distances and the Hubble constant. JWST offers a resolution three times that of the Hubble Space Telescope (HST) with nearly 10 times the sensitivity. It has been suggested that the discrepancy in the value of the Hubble constant based on Cepheids compared to that inferred from measurements of the cosmic microwave background requires new and additional physics beyond the standard cosmological model. JWST observations will be critical in reducing remaining systematics in the Cepheid measurements and for confirming if new physics is indeed required. Early JWST data for the galaxy, NGC 7250 show a decrease in scatter in the Cepheid Leavitt law by a factor of two relative to existing HST data and demonstrate that crowding/blending effects are a significant issue in a galaxy as close as 20 Mpc.

We review the local determination of the Hubble constant, H0, focusing on recent measurements of a distance ladder constructed from geometry, Cepheid variables and Type Ia supernovae (SNe Ia). We explain in some detail the components of the ladder: (1) geometry from Milky Way parallaxes, masers in NGC 4258 and detached eclipsing binaries in the Large Magellanic Cloud; (2) measurements of Cepheids with the Hubble Space Telescope (HST) in these anchors and in the hosts of 42 SNe Ia; and (3) SNe Ia in the Hubble flow. Great attention to negating systematic uncertainties through the use of differential measurements is reviewed. A wide array of tests are discussed. The measurements provide a strong indication of a discrepancy between the local measure of H0 and its value predicted by Λ Cold Dark Matter theory, calibrated by the cosmic microwave background (Planck), a decade-long challenge known as the ‘Hubble Tension’. We present new measurements with the James Webb Space Telescope of >320 Cepheids on both rungs of the distance ladder, in a SN Ia host and the geometric calibrator NGC 4258, showing good agreement with the same as measured with HST. This provides strong evidence that systematic errors in HST Cepheid photometry do not play a significant role in the present Hubble Tension. Future measurements are expected to refine the local determination of the Hubble constant.

The luminosity of the brightest stars (tip) of the red giant branch (TRGB) in the color–magnitude diagrams of old stars was used early on to introduce the ‘multiple stellar populations’ concept, in 1944, by Walter Baade. However, the precision and accuracy of the TRGB for distance estimation has not been known well for long. In the modern era, equipped with high spatial resolution imaging telescopes, the TRGB is considered an excellent standard candle for any type of resolved galaxies, thus representing a powerful probe for cosmology. The TRGB has several advantages over the classical Cepheids. I review how we can apply the TRGB in cosmology. Four science cases, from large to small scales, are presented: (1) the Hubble flow with Type Ia supernovae; (2) Virgo Cluster infall and dark matter; (3) dark galaxies; and (4) dark matter-free galaxies.

Local Group galaxies, particularly the Large and Small Magellanic Clouds, have historically played and continue to play a unique role in studies of the period–luminosity (PL), period–luminosity–color (PLC), and period–Wesenheit (PW) relations, not just for pulsating stars. In recent years, significant efforts have been devoted to calibrate the PL, PLC, and PW relationships at different wavelengths, including studies of the influence of metallicity and nonlinearities on the accuracy of measured distances. However, the PL diagram has many more astrophysical applications. It serves as a vital tool for classifying different types of pulsating stars and can even facilitate the discovery of new classes of variable stars. Moreover, it aids in distinguishing among various modes of pulsation, facilitates the identification of pulsating stars that are members of binary systems, and enables studies of the three-dimensional structures of neighboring galaxies. In this contribution, I present the latest results on the PL, PLC, and PW relations obeyed by various types of variable stars in Local Group galaxies – from δ Scuti stars to Mira variabless and from close binary systems to the mysterious long secondary periods exhibited by red giant and supergiant stars.

Pulsating stars play a crucial role in the calibration of the cosmic distance scale as well as in tracing the properties of the associated stellar populations. In the era of large observational surveys and precise astrometric missions, it is crucial to rely on accurate stellar pulsation models able to predict the observed behaviors for different physical assumptions. Indeed, the relations currently used in the literature to derive individual and mean distances of mainly radially pulsating stars such as Cepheids and RR Lyrae are well physically understood, but are also known to depend on a number of often unknown parameters. Recent extensive sets of stellar pulsation models developed by various authors show how variations in the physical assumptions can affect the theoretical prediction of the instability strip boundaries, the morphology and amplitude of light and radial velocity curves, and the consequent Period-Luminosity, Period-Luminosity-Color and Period-Wesenheit relations. These aspects are discussed in the framework of current open problems in the field of classical pulsating stars.

In the context of a project aimed to provide an updated theoretical scenario for various classes of radially pulsating stars, we present the first results obtained for anomalous Cepheids. By adopting reliable and updated evolutionary prescriptions concerning the luminosity levels for various core He-burning stellar models with masses suitable for entering the instability strip, we have computed nonlinear convective pulsation models for both fundamental and first-overtone mode anomalous Cepheids by exploring the impact of varying the metal abundance as well as the efficiency of super-adiabatic convection. These numerical simulations have allowed us to retrieve the boundaries of the instability strip and all relevant pulsation properties, namely period, amplitude, bolometric light and radial velocity curves. This theoretical scenario has been transformed into the Gaia photometric system to derive the first theoretical Period–Luminosity–Colour and Period–Wesenheit relations in the Gaia bands.

This paper reviews our current knowledge about pulsating chemically peculiar (CP) stars. CP stars are slowly rotating upper main-sequence objects, efficiently employing diffusion in their atmospheres. They can be divided into magnetic and non-magnetic objects. Magnetic activity significantly influence their pulsational characteristics. Only a handful of magnetic, classical pulsating objects are now known. The only exceptions are about 70 rapidly oscillating Ap stars, which seem to be located within a very tight astrophysical parameter space. Still, many observational and theoretical efforts are needed to understand all important physical aspects and their interrelationships. The most important steps to reach these goals are reviewed.

Pulsating variable δ Scuti stars are intermediate-mass stars with masses in the range of 1–3 δ and spectral types between A2 and F2. They can be found at the intersection of the Cepheid instability strip with the main sequence. They can be used as astrophysical laboratories to test theories of stellar evolution and pulsation. In this contribution, we investigate the observed period–colour and amplitude–colour (PCAC) relations at maximum/mean/minimum light of Galactic bulge and Large Magellanic Cloud δ Scuti stars for the first time and test the hydrogen ionization front (HIF)-photosphere interaction theory using the mesa-rsp code. The PCAC relations, as a function of pulsation phase, are crucial probes of the structure of the outer stellar envelope and provide insight into the physics of stellar pulsation and evolution. The observed behaviour of the δ Scuti PCAC relations is consistent with the theory of the interaction between the HIF and the stellar photosphere.

We present new theoretical period–luminosity (PL) and period–radius (PR) relations at multiple wavelengths (Johnson–Cousins–Glass and Gaia passbands) for a fine grid of BL Herculis models computed using mesa-rsp. The non-linear models were computed for periods typical of BL Her stars, i.e. 1 ≤ P(days) ≤ 4, covering a wide range of input parameters: metallicity (−2.0 dex ≤ [Fe/H] ≤ 0.0 dex), stellar mass (0.5–0.8 ), luminosity (50–300 ) and effective temperature (full extent of the instability strip; in steps of 50K). We investigate the impact of four sets of convection parameters on multi-wavelength properties. Most empirical relations match well with theoretical relations from the BL Her models computed using the four sets of convection parameters. No significant metallicity effects are seen in the PR relations. Another important result from our grid of BL Her models is that it supports combining PL relations of RR Lyrae and Type II Cepheids together as an alternative to classical Cepheids for the extragalactic distance scale calibration.

We discuss the impact of Gaia, the cornerstone mission of the European Space Agency (ESA), on the calibration of the period–luminosity and luminosity–metallicity relations of Cepheids and RR Lyrae stars, with specific reference to data published as part of the most recent Gaia releases: Early Data Release 3 (EDR3), on 19 December 2020, and Data Release 3 (DR3) on 13 June 2022. We provide future perspectives for the Gaia mission, including extensions approved by ESA and a tentative schedule of the data releases that will take place in the next few years. We briefly present plans for cross-Coordination Unit processing of Gaia data of Cepheids and RR Lyrae stars for DR4 and conclude by outlining the expected improvement in astrometry at the end of the extended Gaia mission, which will help to further strengthen the calibration of the first rung of the cosmic distance ladder.

In this invited review I discuss the calibration and applications of the period–luminosity relation of classical Cepheid and RR Lyrae stars. After a brief introduction, starting with results from Hipparcos and discussing some post-Hipparcos era developments, I focus on recent results using Gaia Data Release 3 data. I present an overview of the most recent period–luminosity relations, a discussion and some new results on Cepheids in open clusters. I also discuss the effect of reddening and that the use of Wesenheit indices is actually an oversimplification to dealing with the problem of reddening.

We present a progress report of our project aiming to increase the number of known Cepheids in double-lined binary (SB2) systems from six to 100 or more. This will allow us, among other goals, to accurately measure masses for a large sample of Cepheids. Currently, only six accurate Cepheid masses are available, which hinders our understanding of their physical properties and renders the Cepheid mass–luminosity relation poorly constrained. At the same time, Cepheids are widely used for essential measurements (e.g., extragalactic distances, the Hubble constant). To examine Cepheid period–luminosity relations, we selected as binary candidates Cepheids that are too bright for their periods. To date, we have confirmed 56 SB2 systems, including the detection of significant orbital motions of the components for 32. We identified systems with orbital periods up to five times shorter than the shortest reported period to date, as well as systems with mass ratios significantly different from unity (suggesting past merger events). Both features are essential to understand how multiplicity affects the formation and destruction of Cepheid progenitors and what effect this has on global Cepheid properties. We also present eight new systems composed of two Cepheids (only one such system was known before). Among confirmed SB2 Cepheids, there are also several wide-orbit systems. In the future, these may facilitate independent accurate geometric distance measurements to the Large and Small Magellanic Clouds.

We present astrometric very long baseline interferometry (VLBI) studies of AGB stars. To understand the properties and evolution of AGB stars, distances are an important parameter. The distribution and kinematics of their circumstellar matter are also revealed with the VLBI method. We used the VERA array to observe 22 GHz H2O masers in various subclasses of AGB stars. Parallaxes of the three OH/IR stars NSV17351, OH39.7+1.5, IRC−30363, and the Mira-type variable star AW Tau were newly obtained. We present the circumstellar distribution and kinematics of H2O masers around NSV17351. The absolute magnitudes in mid-infrared bands of OH/IR stars with very long pulsation periods were investigated and a period-magnitude relation in the WISE W3 band, MW3 = (−7.21 ± 1.18) log P + (9.25 ± 3.09), was found for the Galactic AGB stars. The VLBI is still a powerful tool for parallax measurements of the Galactic AGB stars surrounded by thick dust shells.

Disentangling the structural components of the Milky Way requires knowledge of distances to various classes of objects, both young, which trace the Galactic disk, and old, which trace the Galactic bulge and halo. Variable stars that obey period-luminosity relations are perfect distance indicators for such studies. Here we discuss recent findings on the structure of our galaxy, inferred from period-luminosity relations for both young, old and intermediate-age variable stars, including Cepheids, RR Lyrae stars, δ Scuti stars, and long-period variables.

The evolution of granulation is an important mechanism of the light variations of red supergiants (RSGs). Based on pure and complete samples of RSGs in the Magellanic Clouds, the mechanisms and characteristics of the granulation of RSGs are investigated based on time-series data. As predicted by the basic physical process of granulation and previous works, there are tight relations between granulation and stellar parameters of RSGs (i.e., the scaling relations). The scaling relations of RSGs provide a new method to infer stellar parameters by using the characteristic timescale and amplitude of granulations. Some faint sources deviate from the scaling relations, which may be due to the difference in the properties of the granulation of the RSGs before and after the blue loop or contamination by Mira variables. However, both of these possibilities suggest that the scaling relations of granulation is different among different types of stars.

We employed data from the VISTA near-infrared YJKS survey of the Magellanic System to analyse the light curves of Type II Cepheids (T2Cs) in the Large and Small Magellanic Clouds (LMC and SMC, respectively). Using the T2Cs identified by the OGLE IV survey and Gaia mission, we built up a sample of about 330 pulsators belonging to both galaxies. For all these objects we obtained accurate intensity-averaged magnitudes in the YJKS bands by means of a template-fitting technique. We complemented our near-infrared data with optical photometry from the literature to calculate period-luminosity and period-Wesenheit relations for a variety of different bands and colour combinations and separately for the different T2C subclasses (BL Herculis, W Virginis, peculiar W Virginis, RV Tauri). These relations, calibrated with the LMC distance modulus, were tested using T2Cs belonging to Galactic globular clusters. We thus calculated the distances of 22 clusters and compared them with the literature values, mainly based on RR Lyrae stars, finding good agreement within 1 σ and dispersion of the order of 0.3 – 0.5 kpc, depending on the adopted period-luminosity/period-Wesenheit relation.

Our knowledge of stellar evolution relies on constraints provided by measurements of the physical stellar properties such as the mass, effective temperature, and radii. The most fundamental parameter, the stellar mass, is rarely available or has a low accuracy, providing poor constraints on the stellar structure and evolution. Observing binary stars combining astrometry and spectroscopy offers the unique opportunity to measure very precise masses. In addition, double-lined spectroscopic binaries provide independent distance measurements with an extreme accuracy, allowing to test the Gaia parallaxes and the period-luminosity (P-L) relations. I will show that masses and distances with an accuracy level as high as 0.05% can be obtained by combining interferometric and spectroscopic observations for different types of binary systems, i.e. binary Cepheids, eclipsing and non-eclipsing binaries.

We present mean horizontal branch absolute magnitudes and iron abundances for a sample of 39 globular clusters. These quantities were calculated in an unprecedented homogeneous fashion based on Fourier decomposition of ligt curves of RR Lyrae cluster members. Zero points for the luminosity calibrations are discussed. Our photometrically derived metallicities and distances compare very well with spectroscopic determinations of [Fe/H] and accurate distances obtained using Gaia and Hubble Space Telescope data. The need to distinguish between the results for RRab and RRc stars for a correct evaluation of the MV–[Fe/H] relation is discussed. For RRab stars, the relation is non-linear, and the horizontal branch structure plays a significant role. For RRc stars, the relation remains linear and tight, and the slope is very shallow. Hence, the RRc stars seem better indicators of the parental cluster distances. Systematic time-series CCD imaging performed over the last 20 years enabled to discover and classify 330 variables in our sample of globular clusters.

In this work, we focus on the period-luminosity relation (PLR) of δ Sct stars, in which mode excitation and selection mechanisms are still poorly constrained, and whose structure and oscillations are affected by rotation. We review the PLRs in the recent literature, and add a new inference from a large sample of δ Sct. We highlight the difficulty in identifying the fundamental mode and show that rotation-induced surface effects can impact the measured luminosities, explaining the broadening of the PLR. We derive a tight relation between the low-order large separation and the fundamental radial mode frequency (F0) that holds for rotating stars, thus paving the way towards mode identification. We show that the PLRs we obtain for different samples are compatible with each other and with the recent literature, and with most observed δ Sct stars when taking rotation effects into account. We also find that the highest-amplitude peak in the frequency spectrum corresponds to the fundamental modein most δ Sct, thus shedding some light on their elusive mode selection mechanism.

Classical pulsating stars such as Cepheid and RR Lyrae variables exhibit well-defined Period–Luminosity relations at near-infrared wavelengths. Despite their extensive use as stellar standard candles, the effects of metallicity on Period–Luminosity relations for these pulsating variables, and in turn, on possible biases in distance determinations, are not well understood. We present ongoing efforts in determining accurate and precise metallicity coefficients of Period–Luminosity-Metallicity relations for classical pulsators at near-infrared wavelengths. For Cepheids, it is crucial to obtain a homogeneous sample of photometric light curves and high-resolution spectra for a wide range of metallicities to empirically determine metallicity coefficient and reconcile differences with the predictions of the theoretical models. For RR Lyrae variables, using their host globular clusters covering a wide range of metallicities, we determined the most precise metallicity coefficient at near-infrared wavelengths, which is in excellent agreement with the predictions of the horizontal branch evolution and stellar pulsation models.