The aim of this chapter is to present a consistent description of the mathematical and numerical investigation on primordial black hole formation from the collapse of adiabatic cosmological perturbations performed in the last 25 years, describing the mechanism of gravitational collapse. Despite the presence of some technicalities, at the end of this chapter the reader will should have a consistent scenario, knowing how the results of numerical simulations depend from the initial conditions described by the power spectrum of cosmological perturbations. This will provide the basic knowledge required to take the first steps in the research of primordial black hole formation.