Abstract: Harish-Chandra has given a simple and explicit classification of the discrete series representations of reductive groups over the real numbers. We will describe a very similar classification that holds for a large proportion of the supercuspidal representations of reductive groups over non-archimedean local fields (which we may call regular). The analogy runs deeper: there is a remarkable parallel between the characters of regular supercuspidal representations and the characters of discrete series representations of real reductive groups. This leads to an explicit construction of the local Langlands correspondence for supercuspidal Langlands parameters (without regularity assumptions), under mild conditions on the residual characteristic. The treatment of the non-regular case presents additional challenges, due to the occurrence of non-abelian centralizer groups on the Galois side.