  Primordial black holes, which could have been formed in the very early
Universe due to the collapse of large curvature fluctuations, are nowadays one
of the most attractive and fascinating research areas in cosmology for their
possible theoretical and observational implications. This review article
presents the current results and developments on the conditions for primordial
black hole formation from the collapse of curvature fluctuations in spherical
symmetry on a Friedman-Lemaitre-Robertson-Walker background and its numerical
simulation. We review the appropriate formalism for the conditions of
primordial black hole formation, and we detail a numerical implementation. We
then focus on different results regarding the threshold and the black hole mass
using different sets of curvature fluctuations. Finally, we present the current
state of analytical estimations for the primordial black hole formation
threshold, contrasted with numerical simulations.
