This open access book offers a concise yet pedagogic introduction to the theory of cosmological inflation and the early epochs that follow it. Inflation explains how the seeds for density perturbations could have formed during a period of exponential expansion. Apart from density perturbations, also so-called tensor perturbations are generated, which may be observed as gravitational waves. Starting from the Einstein equations of general relativity, the formalism is developed through explicit computations and a careful explanation of both the mathematics and the physics involved, paying special attention to general-relativistic gauge invariance. For technically cumbersome steps, examples of numerical and symbolic computer scripts are provided. We also discuss the issue of thermalization, by which a part of the energy density driving the exponential expansion can eventually be converted into the conventional hot big bang. The book is primarily aimed at graduate students of theoretical high-energy physics, but can serve as a self-contained introduction to inflation also for advanced undergraduate students, or for professional scientists.