APPROVED In this thesis, we study the modularity and duality of Coulomb branches for a class of four-dimensional N=2 supersymmetric gauge theories. For pure N=2 super Yang-Mills theory with gauge group SU(2), the Coulomb branch can be identified with the modular fundamental domain for the duality group of the theory. We extend this result for higher rank gauge groups and for the inclusion of matter. In N=2 supersymmetric quantum chromodynamics, we find a completely general description in terms of non-modular fundamental domains for the effective gauge coupling. For the superconformal SU(2) theory with four flavours, the order parameters in certain cases enhance to bimodular forms. When the gauge group is enlarged to SU(3), the two-dimensional space of vacua contains natural one-dimensional elliptic loci which admit a modular parametrisation. For the topologically twisted theories on a four-manifold, these fundamental domains are natural integration domains for topological correlation functions. We study the formulation and evaluation of such correlation functions for a class of N=2 supersymmetric gauge theories.