Since the time of Friedman, Lemaitre, Gamow, and Hoyle, and the discovery of the cosmological microwave background radiation by Penzias and Wilson, we know that the very early state of the universe must have been extremely hot and dense. The universe must have originated from a very small initial state in what is called the Big Bang. However, the conventional hot Big Bang has several inconsistencies in that it requires extremely fine-tuned initial conditions. These problems have so far been most successfully addressed in the scenario of cosmological inflation. A very early period of hyperfast exponential expansion solves the initial condition problems of the hot Big Bang. In addition it provides a viable origin of the large-scale structure observed in the universe. We will discuss the basic structure and predictions of cosmological inflation. We will also describe an inevitable feature of inflation, namely eternal inflation - an inflating universe will never completely stop inflating, and there will be regions where inflation will go on forever. When we try to embed inflation into a candidate theory of quantum gravity, string theory, we have to confront recent results pointing to an unfathomably large 'landscape' of possibly more than 10^1000 different solutions (vacua). We will discuss the origin of this extremely large multitude of string theory vacua. When we combine this with inflation, we see that the presence of the string theory landscape and eternal inflation together implies the presence of an infinitely large multiverse which realizes all possible solutions of string theory infinitely often.