Aspects of supersymmetric gauge theories in two and three dimensions

In this thesis we study two- and three-dimensional supersymmetric gauge theories, in particular 2d $\mathcal{N}=(2,2)$ and 3d $\mathcal{N}=4$ theories. The techniques of supersymmetric localization and the Jeffrey-Kirwan residue are applied to compute correlation functions in these theories. Using the localization result for the correlation functions of 2d $\mathcal{N}=(2,2)$ Gauged Linear Sigma Models (GLSMs) on the Omega-deformed two-sphere, we examine the correlation functions of a GLSM describing a non-compact geometry. We investigate the ambiguity in the results for three-point correlators using twisted masses and the Omega deformation, and we compare with previous evaluations of these correlation functions in the literature. For 3d $\mathcal{N}=4$ $U(N)$ theories, by combining inputs from supersymmetric localization and brane constructions in type IIB string theory, we compute correlation functions of monopole operators that are inserted in an Omega background. We study various examples of correlators involving the product of monopoles of minimal positive and negative charges, and investigate the effects of monopole bubbling and wall-crossing phenomena. Our results are successfully tested using the non-commutative Moyal (star) product and the action of Parity-Time (PT) symmetry.