Abstract: In his treatise on "Symmetry", Hermann Weyl credits Leonardo Da Vinci with the insight that the only finite symmetry groups in the plane are cyclic or dihedral. Reaching back even farther, had the abstract notion been around, Euclid's Elements may well have ended with the theorem that only three further groups can occur as finite groups of rotational symmetries in 3-space, namely those of the Platonic solids. Of course, it took another 22 centuries for such a formulation to be possible, put forward by C.Jordan (1877) and F.Klein (1884). The most accessible approach is due to Coxeter (1934) who first classified the finite symmetry groups generated by reflections. Especially Klein's investigation of the orbit spaces of those groups and their double covers, the binary polyhedral groups, is at the origin of singularity theory and in the century afterwards many surprising connections with other areas of mathematics such as the theory of simple Lie groups were revealed in work by Grothendieck, Rieskorn, and Slodowy in the 1960's and 70's. It came then as a complete surprise when J.McKay pointed out in 1979 a very direct, though then mysterious relationship between the geometry of the resolution of singularities of these orbit spaces and the representation theory of the finite groups one starts from. In particular, he found a simple explanation for the occurrence of the Coxeter-Dynkin diagrams in the theory. This marks essentially the beginning of “Noncommutative Singularity Theory", a subject area that has exploded during the last decade in particular because of its role in the mathematical formulation of String Theory in Physics. In this talk I will survey the beautiful classical mathematics at the origin of this story and then give a sampling of recent results and work still to be done.