  We present a nonlinear study of the inflationary epoch based on numerical
lattice simulations. Lattice simulations are a well-known tool in primordial
cosmology, and they have been extensively used to study the reheating epoch
after inflation. We generalize this known machinery to the inflationary epoch.
Being this the first simulation of the inflationary epoch much before the end
of inflation, the first part of the thesis focuses on the minimal single-field
model of inflation. We discuss the conceptual and technical ingredients needed
to simulate inflation on a lattice. The simulation is used to reproduce the
nearly scale-invariant spectrum of scalar perturbations, as well as the
oscillations in the power spectrum caused by a step in the potential. In the
second part, we focus on the more complicated axion-U(1) model of inflation and
present the first lattice simulation of this model during the deep inflationary
epoch. We use the simulation to discover new properties of primordial scalar
perturbations from this model. In the linear regime of the theory, we find
high-order non-Gaussianity (beyond trispectrum) to be key to describing the
statistical properties of scalar perturbations. Conversely, we find
perturbations to be nearly Gaussian in the nonlinear regime of the theory. This
relaxes existing constraints from the overproduction of primordial black holes,
allowing for a gravitational waves signal in the observable range of upcoming
experiments such as LISA. Our results show that lattice simulations can be a
powerful tool to study the inflationary epoch and its observational signatures.
