Cosmic inflation is arguably the most favoured paradigm of the very early Universe. It postulates an early phase of fast, nearly exponential, and accelerated expansion. Inflationary models are capable of explaining the overall flatness and homogeneity of today's Universe at large spatial scales. Furthermore, their predictions give the observed statistical features of the primordial density fluctuations imprinted in the cosmic microwave background and the Universe's large scale structure. However, despite the wide acceptance in the physics community, these models are not absent from criticism. In scalar field inflation, a necessary condition to begin inflation is the requirement of a Universe dominated by the field's potential, which implies a subdominant contribution from the scalar field dynamics, i.e. gradients and field velocities. This has given rise to large amounts of scientific debate and literature on the naturalness, and possible fine-tuning of the initial conditions for inflation. Another controversial issue concerns the ending of inflation, and the fact that a preheating mechanism is necessary to transfer the energy of the inflaton to the bath of highly energetic particles of the hot big bang plasma. In this thesis, we present full general relativistic simulations to study these two problems, with a particular focus on the Starobinsky and Higgs models of inflation, being those the most favoured by the latest observations. First, we consider the fine-tuning problem of beginning inflation from a highly dynamical and inhomogeneous "preinflation" epoch in the single-field case. In our second study, we approach the multifield paradigm of inhomogeneous preinflation, together and consistently, with the preheating phase after inflation. These investigations further confirm the robustness of these types of models to highly inhomogeneous initial conditions, while putting in evidence the non-negligible effects of gravitational effects during the (p)reheating. At the end of the manuscript, we finalize by discussing some other potential applications of numerical simulations to study the early Universe, including our preliminary investigations on primordial black hole formation in asymmetric three-dimensional configurations.