In this talk, I will focus on the behavior of matter at high densities in compact stars and the challenges that emerged in recent years in the nuclear physics and astrophysics of these objects. The possibility of hyperonization of dense baryonic matter, the so-called „hyperon puzzle“ and its solutions within the covariant density functional theories of hypernuclear matter will be discussed. I will then go on to discuss the phase transition from baryonic to deconfined quark matter and its consequences for compact star astrophysics. In particular, I will show that a strong first-order phase transition among the phases of dense quark matter can lead to twin and triplet configurations of compact stars, i.e., stars with same masses but different radii. Neutrinos and axions are the main cooling agents that determine the thermal evolution of compact stars during their first ten thousand years. I will discuss the methods for computing of the rates of these processes and their impact on cooling simulations of compact stars. Finally, I will discuss the implications of the recent detection of gravitational waves from a neutron star binary inspiral for the tidal deformability of compact stars and the equation of state of dense matter.