Simple Singularities and N=2 Supersymmetric Yang-Mills Theory

We present a first step towards generalizing the work of Seiberg and Witten on N=2 supersymmetric Yang-Mills theory to arbitrary gauge groups. Specifically, we propose a particular sequence of hyperelliptic genus $n-1$ Riemann surfaces to underly the quantum moduli space of $SU(n)$ N=2 supersymmetric gauge theory. These curves have an obvious generalization to arbitrary simply laced gauge groups, which involves the A-D-E type simple singularities. To support our proposal, we argue that the monodromy in the semiclassical regime is correctly reproduced. We also give some remarks on a possible relation to string theory.