We study theoretical aspects of the early Universe, with a particular focus on primordial black holes (PBHs). PBHs are black holes that may have formed in the early Universe through several mechanisms, most notably the gravitational collapse of rare, large-amplitude density perturbations. They provide a unique probe of small-scale primordial fluctuations, with implications for dark matter, gravitational-wave signatures, and high-energy physics. Our goal is to understand the conditions for PBH formation, their abundance and mass function, and the observational signatures they may produce. To this end, we combine analytical methods, statistical descriptions of primordial initial conditions, and numerical simulations, including relativistic simulations of gravitational collapse.