We present a comprehensive description of the population synthesis code StarTrack. The original code has been significantly modified and updated. Special emphasis is placed here on processes leading to the formation and further evolution of compact objects (white dwarfs, neutron stars, and black holes). Both single and binary star populations are considered. The code now incorporates detailed calculations of all mass transfer phases, a full implementation of orbital evolution due to tides, as well as the most recent estimates of magnetic braking. This updated version of StarTrack can be used for a wide variety of problems, with relevance to observations with many current and planned observatories, e.g., studies of X-ray binaries (Chandra, XMM-Newton), gravitational radiation sources (LIGO, LISA), and gamma-ray burst progenitors (HETE-II, Swift). The code has already been used in studies of Galactic and extragalactic X-ray binary populations, black holes in young star clusters, Type Ia supernova progenitors, and double compact object populations. Here we describe in detail the input physics, we present the code calibration and tests, and we outline our current studies in the context of X-ray binary populations.