I will discuss groups generated by reflections on the spaces on constant curvature: the sphere, Euclidean space, or hyperbolic space. Then I will define the notion of a "Coxeter group" (or an "abstract reflection group"). I will show how to realize any given Coxeter group as a group generated by reflections on a particularly nice space. In the second talk I will give applications of this construction to some problems in geometric group theory and topology.