Starting with Lovász' solution of Kneser's conjecture in 1978, a number of problems in discrete mathematics and theoretical computer science have been solved using methods from algebraic and geometric topology. The aim of this course is to explain some of these results and methods to a broad audience; the precise choice of topics will depend on the audience's background; sample topics incude: Topological lower bounds for the chromatic number of a graph Decision tree complexity and evasiveness of graph properties Nonembeddability and Tverberg-type results Equipartition results for mass distributions Matchings in hypergraphs Impossibility theorems in distributed computing