The KKM theorem, due to Knaster, Kuratowski and Mazurkiewicz in 1929, is a topological lemma reminiscent of Sperner's lemma and Brouwer's fixed point theorem. It has numerous applications in combinatorics, discrete geometry, economics, game theory and other areas. Generalizations of this lemma, in several different directions, were proved over the years (e.g., by Shapley, Gale, Shih-Lee, Komiya, Frick-Zerbib, and Soberon) and have been widely applied as well. In my lecture series we will prove several different KKM-type theorems and use them to solve a variety of problems in discrete geometry, combinatorics and game theory. We will also discuss open problems that may benefit from a similar topological approach.