After a short description of the historical discovery of pulsars, the different possible interpretations for pulsars are discussed. The different observational classes of pulsars are summarized. The dipole model for pulsars is introduced to motivate the definition of the characteristic age and the magnetic field of pulsars. The notion of the braking index of pulsars is confronted with the dipole model and the emission of gravitational waves from pulsars. The pulsar diagram is discussed in terms of the evolution of pulsars and the recycling mechanism for binary pulsars. The model of the aligned rotator introduces the magnetosphere of pulsars and a more sophisticated approach for the pulsar emission mechanism. The details of extracting the neutron star masses from pulsar timing is outlined. Observational data from the Hulse–Taylor pulsar and the double pulsar are confronted with predictions from general relativity in strong fields.

The theory of quantum chromodynamics (QCD) is introduced. Features of QCD as the nontrivial vacuum due to quark and gluon condensate and asymptotic freedom at high-energy scales are discussed. The concept of perturbative QCD and the running of the coupling constant is established. The equation of state of QCD at high temperatures from lattice QCD is reviewed and confronted with perturbative QCD calculations. The QCD equation of state at high baryon density is discussed. Properties of selfbound stars are developed where the equation of state has a nonvanishing pressure at a nonvanishing energy density. The mass–radius relation of pure quark stars is examined and compared to the limits from causality.

This chapter introduces compact objects, white dwarfs, neutron stars, and black holes. The properties of compact objects are summarized as the typical radius, mass, and compactness. These compact objects are the final end point in stellar evolution. The stellar evolution in terms of the mass of the star is outlined, focusing on the burning stages and the final collapse of the star, either as a white dwarf or in a core-collapse supernova. Historical notes are given for the discovery of white dwarfs and neutron stars.

The thermodynamics potentials for describing matter at nonzero temperatures and densities or chemical potentials are summarized. Emphasis is put on the thermodynamically correct description within the canonical and grand canonical ensemble for dense matter. The notion of chemical equilibrium is introduced for several conserved quantities and used to describe matter in β-equilibrium where charge and baryon number are conserved. The limit for nonrelativistic and relativistic particles is worked out in detail. The concept of an equation of state is introduced and applied to free Fermi gases. The pressure integral is solved analytically and the nonrelativistic and relativistic limits for the equation of state are delineated. Finally, the properties of polytropes are discussed and connected to the limiting cases of the equation of state of a free Fermi gas.

As one of the core chapters, the general properties of compact stars are discussed. Spheres of fluid in hydrostatic equilibrium are studied within general relativity. The concept of the mass–radius relation is introduced for the classic case of a gas of noninteracting neutrons. Landau‘s argument for a maximum mass of neutron stars and white dwarfs is delineated. Thereby, the Landau mass and radius is defined for studying scaling solutions of the Tolman–Oppenheimer–Volkoff equation. The power of scaling arguments is demonstrated for the case of a free Fermi gas with arbitrary particle mass, a relativistic gas of fermions with a vacuum term, and the limiting equation of state from causality. The concept of selfbound stars is put forward, giving rise to limits on the compactness and the maximum density achievable for compact stars in general. Generic interactions between fermions are studied and their implications for compact star properties are derived. The general properties of compact stars made of bosons with and without interactions are also investigated.

Hybrid stars are constructed by combining the neutron matter equation of state with the quark matter equation of state. Piecewise polytropes are utilized to interpolate between the limiting equations of state at low and high densities. The phase diagram of QCD is sketched and possible phase transitions are outlined. Connections to heavy-ion experiments and astrophysical systems are established. The two possible ways of constructing a phase transition, the Maxwell and the Gibbs construction, are developed. The possible existence of a pasta phase in the core is presented. The implications of a phase transition for the mass–radius relation of compact stars is discussed. The emergence of a new class of stable solutions of the Tolman–Oppenheimer–Volkoff equation, a third family besides white dwarfs and neutron stars, is put forward. The Seidov criterion for an unstable branch in the mass–radius relation is derived explicitly. Hybrid stars are classified according to the presence of the mixed phase in the mass–radius relation. The properties of hybrid stars are compared to those of ordinary neutron stars by their possible configurations in the mass–radius diagram.