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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 scales. Despite being widely accepted by 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. This has originated 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 end of inflation, and the fact that a preheating mechanism is necessary to originate the hot big bang plasma after inflation. 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. 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 preinflation, together and consistently, with the preheating phase. These investigations confirm the robustness of these inflationary models to generic initial conditions, while putting in evidence the non-negligible gravitational effects during preheating. At the end of the manuscript, we discuss potential applications of numerical simulations in cosmology, including our preliminary investigations on primordial black hole formation.