The simplest form of the infinite Ramsey theorem states that any colouring of pairs of integers in two colours has an infinite monochromatic set. We discuss generalisations to colourings of the set of finite subsets of the integers (Hindman's theorem) and also to colourings of the set of infinite subsets of the integers (the Galvin-Prikry theorem). Topological ideas are central to the proofs of these general Ramsey-type theorems.