The talk will summarize how inflation works, and also the key pieces of evidence that our universe actually underwent this process. Almost all versions of inflation lead to eternal inflation: inflation goes on forever, producing a "multiverse" of "pocket universes". Although the line of reasoning is controverial, such a multiverse might help to explain one of the biggest mysteries in cosmology: the question of why the vacuum energy density (or cosmological constant) is many orders of magnitude smaller than would be expected from our understanding of particle physics.