Szemerédi's regularity lemma is one of the most powerful tools in graph theory, with many applications in combinatorics, number theory, discrete geometry, and theoretical computer science. Roughly speaking, it says that every large graph can be partitioned into a small number of parts such that the bipartite subgraph between almost all pairs of parts is random-like. Several variants of the regularity lemma have since been established yielding many further applications. In this talk, I will survey this area, including recent progress on quantitative estimates in the various regularity lemmas, and alternative methods yielding new proofs of applications with improved quantitative estimates.