Topology provides a bridge between the finite nature of computers and the infinite nature of mathematical objects we want to compute with. The first part of the talk will review the history and main contributions of various protagonists, building on the pioneering insights of Brouwer, Kleene, Kreisel, Myhill & Shepherdson, Scott, Smyth, Abramsky, Vickers, among others. The second part of the talk will discuss my own work, old and recent, including the development of topological ideas in type theory, which serves both as a programming language and as a foundation of constructive mathematics. The talk will be addressed to a general audience, and will not assume familiarity with topology or type theory.