After the work of Seiberg and Witten, it has been seen that the dynamics of N=2 Yang-Mills theory is governed by a Riemann surface \(\Sigma\). In particular, the integral of a special differential \(\lambda_{SW}\) over (a subset of) the periods of \(\Sigma\) gives the mass formula for BPS-saturated states. We show that, for each simple group \(G\), the Riemann surface is a spectral curve of the periodic Toda lattice for the dual group, \(G^\vee\), whose affine Dynkin diagram is the dual of that of \(G\). This curve is not unique, rather it depends on the choice of a representation \(\rho\) of \(G^\vee\); however, different choices of \(\rho\) lead to equivalent constructions. The Seiberg-Witten differential \(\lambda_{SW}\) is naturally expressed in Toda variables, and the N=2 Yang-Mills pre-potential is the free energy of a topological field theory defined by the data \(\Sigma_{\gg,\rho}\) and \(\lambda_{SW}\).