**Abstract:** We will give a gentle introduction to a recently-developed technique for bounding the number (and controlling the typical structure) of finite objects with forbidden substructures. This technique exploits a subtle clustering phenomenon exhibited by the independent sets of uniform hypergraphs whose edges are sufficiently evenly distributed; more precisely, it provides a relatively small family of 'containers' for the independent sets, each of which contains few edges. In the first half of the talk we will attempt to convey a general high-level overview of the method; in the second, we will describe a few illustrative applications in areas such as extremal graph theory, Ramsey theory, additive combinatorics, and discrete geometry. Note that it is a "repetition of the ICM 2018 talk", hence it will have overlap with several previous (seminar) talks, and no new result will be presented.