Singularities offer the opportunity to study some interesting mathematics from a relatively simple point of view. First I'll describe what an affine variety is and what it means for one to have a singularity. Then, I'll discuss the work of Milnor, Brieskorn and others on the topology of a complex hypersurface near one of its singular points, which has connections to knot theory and exotic spheres. If I have time, I'll talk about a special class of singularities coming from invariant theory that were discovered by Klein, and the McKay correspondence, which gives an unexpected connection between resolutions of singularities and the Dynkin diagrams used to classify compact Lie groups.