Supersymmetric indices for sigma models are known to compute topological invariants of the target space on which the sigma model is built. In the case where the target space is a K3 surface, the worldsheet of the sigma model enjoys an N=4 superconformal symmetry. A supersymmetric index known as the elliptic genus can be constructed for this theory and decomposed into a sum of massless and massive characters of the N=4 superconformal algebra governing the symmetries. This index exhibits a phenomenon known as Mathieu moonshine, in which the coefficients of the massive characters in that decomposition are dimensions of representations of the sporadic group Mathieu 24. In this thesis, motivated by this moonshine phenomenon for theories with N=4 superconformal symmetries, we consider sigma models which exhibit a larger N=4 superconformal symmetry on the worldsheet, and discuss two supersymmetric indices which could be applied to such sigma models in search of a new moonshine. We discuss the states which contribute to these indices and calculate one of them for some specific theories.