In this chapter, we describe our recent work on the analytic foundations for the study of degenerate diffusion equations which arise as the infinite population limits of standard models in population genetics. Our principal results concern existence, uniqueness, and regularity of solutions when the data belong to anisotropic Holder spaces, adapted to the degeneracy of these operators. These results suffice to prove the existence of a strongly continuous \({\mathcal{C}}^{0}\)-semigroup. The details of the definitions and complete proofs of these results can be found in [8].