The stellar Initial Mass Function (IMF) can be conveniently represented as a canonical two-part power law function and is largely invariant for star formation regions evident in the Local Group of galaxies. The lack of massive stars in regions of low star formation density and other evidence imply that the IMF is not a probability density distribution function, but is instead closer to an optimal density distribution function. Binary stars and stellar dynamics have a very significant influence on the counts of low-mass stars and need to be corrected for. This chapter offers a review of how recent advancements in the measurement of the IMF from detailed star counts in different environments (e.g., young massive clusters, globular clusters, elliptical galaxies and ultrafaint dwarf galaxies), properly interpreted via detailed numerical models, indicate that the environmental conditions and metallicity both have an impact on the shape of the IMF.

The so-called blue straggler Ssars (BSSs) represent the most striking (and certainly the most famous) evidence of binary stars in Galactic Globular Clusters (GCs). In this chapter, the most intriguing properties of BSSs are discussed and two innovative tools based on the physical properties of these fascinating objects are presented: (1) the dynamical clock, and (2) the stellar scale. The former uses the level of central segregation of BSSs to empirically measure the level of dynamical evolution suffered by the parent cluster. The the stellar scale is a spectroscopic tool that allows differential measures of stellar mass able to efficiently distinguish massive objects (as the elusive evolved BSS progeny) from normal low-mass cluster stars.

We still do not have an end-to-end theory of binary star formation that both satisfies observational constraints and also includes all necessary physical ingredients. Large-scale star formation simulations do an excellent job of replicating binary statistics under severely simplified physical conditions (neglect of thermal feedback and magnetic fields). Simulations that include these processes, however, tend to suppress binary formation, and their extra computational expense makes it hard to generate statistical samples of binaries for observational comparison. In addition to reviewing the literature on binary formation simulations, this chapter also examines the insights into the process that are provided by observations of the youngest protomultiple systems.

The binary fraction of metal-poor stars provides important constraints on star formation in the early Galaxy, and is a key piece of information in the understanding the origin of the observed high frequency of C enhanced metal-poor stars. It is now widely accepted that a majority of solar metallicity stars are in binaries; it is not clear, however, if this is the case for metal-poor stars. While state-of-the-art models agree in predicting an increase in the binary fraction and a shift towards lower values for the orbital period distribution at extremely low metallicities, the observational findings paint a patchier picture. This chapter summarises the key motivations for the study of binaries in the very metal-poor regime and reviews the current state of the field and the plans for the future.

In this chapter, the focus is on the properties of post–Asymptotic Giant Branch (post-AGB) stars in binary systems. Their spectral energy distributions (SEDs) are very characteristic: they show a near-infrared excess, indicative of the presence of warm dust, while the central stars are too hot to be in a dust-production evolutionary phase. This allows for an efficient detection of binary post-AGB candidates. It is now well established that the near-infrared excess is produced by the inner rim of a stable dusty disc that surrounds the binary system. These discs are scaled-up versions of protoplanetary discs and form a second generation of stable Keplerian discs. They are likely formed during a binary interaction process when the primary was on ascending the AGB. The chapter summarises what has been learnt so far from the observational properties of these post-AGB binaries. The impact of the creation, lifetime and evolution of the circumbinary discs on the evolution of the system is yet to be fully understood.

Short-duration gamma-ray bursts (short-GRBs) are thought to be produced during the merger ofcompact binary stars involving at least one neutron star. The recent detection of a gravitational wave signal coincident with a short-GRB (170817), albeit one with unusually low intrinsic luminosity, has cemented this link and opened a new era of multimessenger astrophysics. Long-duration gamma-ray bursts are produced by the core collapse of envelope-stripped massive stars, which may also be the end product of binary evolution. Establishing the nature of the long-GRB progenitor more definitely is important not only for our understanding of GRBs, but also for their use as probes of the distant Universe, many of which depend on how representative GRBs are of the general population of massive stars.

The observational parameter space that allows us to detect and describe nonsingle stars is enormous. It comes from the fact that binary stars are very numerous, present themselves with a huge variety of physical properties and have signatures in all astronomical fundamental techniques (astrometry, photometry, spectroscopy). It is, therefore, not a surprise that any significant improvement in observational astronomical facilities has an important impact on our knowledge of binaries. We are currently in an era where the development of various large-scale surveys is impressive. Among them, Gaia and LSST are exceptional surveys that have and likely will have a profound and long-lasting impact on the astronomical landscape. This chapter reviews the status of these two projects, and considers how they improve our knowledge of binary stars.

The chapter presents a summary of the present-day understanding of Type Ia supernova progenitors, mostly discussing the observational approach. This chapter is to provide the nonspecialist with a sufficiently comprehensive view of where we stand.

The statistical distributions of main-sequence multiple-star properties reveal invaluable insights into the processes of binary star formation, and they provide initial conditions for population synthesis studies of binary star evolution. Binary stars are discovered and characterised through a variety of techniques. Correcting for their respective selection effects and combining the bias-corrected results is not a trivial process. This is partially because the intrinsic distributions of companion frequency, primary mass M1, orbital period P, mass ratio q and eccentricity e are all interrelated , i.e., f(M1,P,q,e)/= f(M1)f(P)f(q)f(e). In particular, the binary fraction increases with primary mass, especially across short orbital periods, and binaries become weighted towards larger eccentricities and more extreme mass ratios with increasing separation, especially for more massive primaries. Moreover, binary star statistics vary with age, environment and metallicity. This chapter summarises the strengths and limitations of the various observational techniques, and reviews the statistical correlations in the intrinsic (bias-corrected) multiple-star properties.

With the discovery of both binary black hole mergers and a binary neutron star merger, the field of gravitational wave astrophysics has really begun. The LIGO and Virgo detectors will soon improve their sensitivity allowing for the detection of thousands new sources. All these measurements will provide new answers to open questions in binary evolution related to mass transfer, out-of-equilibrium stars and the role of metallicity. The data will give new constraints on uncertainties in the evolution of (massive) stars, such as stellar winds, the role of rotation and the final collapse to a neutron star or black hole. In the long run, the thousands of detections by the Einstein Telescope will enable us to probe their population in great detail over the history of the Universe. For neutron stars, the first question is whether the first detection GW170817 is a typical source or not. In any case, it has spectacularly shown the promise of complementary electromagnetic follow-up. For white dwarfs, we have to wait for LISA (around 2034), but new detections by, e.g., Gaia and LSST will prepare for the astrophysical exploitation of the LISA measurements.

This chapter discusses the population and spectral synthesis of stellar populations. It describes the method required to achieve such synthesis and discusses examples where inclusion of interacting binaries are vital to reproducing the properties of observed stellar systems. These examples include the Hertzsprung–Russel diagram, massive star number counts, core-collapse supernovae and the ionising radiation from stellar populations that power both nearby HII regions and the epoch of reionization. It finally offers some speculations on the future paths of research in spectral synthesis.

With stellar masses in the range of eight to several hundreds of solar masses, massive stars are among the most important cosmic engines. Each individual object strongly impact its local environment, and entire populations of massive stars have been driving the evolution of galaxies throughout the history of the Universe. Over the last two decades, it has become increasingly clear that massive stars do not form nor live in isolation but rather as part of a binary or higher-order multiple system. Understanding the life cycle of massive multiple systems, from their birth to their death as supernovae and long-duration gamma ray bursts, is thus one of the most pressing scientific endeavours in modern astrophysics. In this quest, observations offer a critical insight that both guide theoretical developments and challenge the model predications. This chapter provides an overview of the observational constraints of the multiplicity properties of OB stars obtained since 2010.

Binaries are the most important energy reservoir of star clusters. Via three-body encounters, binaries can reverse the core collapse and prevent a star cluster from reaching equiparition. Moreover, binaries are essential for the formation of stellar exotica, such as blue straggler stars, intermediate-mass black holes and massive black hole binaries.

Color-magnitude diagrams of open clusters reveal many stars that do not fall on cluster main sequences or red giant branches including blue straggler stars, yellow giants, and sub-subgiants. In fact, as many as a quarter of the evolved stars in older open clusters do not fall on standard single-star isochrones. Rather than being anomalies, these stars are following frequently travelled alternative paths of stellar evolution. Most of these stars are in binary systems, and their origins likely stem from mass transfer, mergers and collisions within binaries. This chapter presents an overview of recent observational and modelling work to understand the processes that shape these alternative stellar evolution pathways, including an HST study of the blue straggler population of NGC 188, an abundance study of the blue stragglers of NGC 6819, establishing yellow giants as evolved blue straggler stars using asteroseismology, exploration of a new class of stars known as sub-subgiants, rotational identification of main sequence blue stragglers with Kepler/K2 and new insights into the angular momentum evolution of blue stragglers.