We explore supersymmetric quantum field theories in three and four dimensions via an analysis of their BPS spectrum. In four dimensions, we develop the theory of BPS quivers which provides a simple picture of BPS states in terms of a set of building block atomic particles, and basic quantum mechanical interactions. We develop efficient techniques, rooted in an understanding of quantum-mechanical dualities, for determining the spectrum of bound states, and apply these techniques to calculate the spectrum in a wide class of field theories including ADE gauge theories with matter, and Argyres-Douglas type theories. Next, we explore the geometric content of quivers in the case when the four-dimensional field theory can be constructed from the six-dimensional (2; 0) superconformal field theory compactified on a Riemann surface. We find that the quiver and its superpotential are determined by an ideal triangulation of the associated Riemann surface. The significance of this triangulation is that it encodes the data of geodesics on the surface which in turn are the geometric realization of supersymmetric particles. Finally we describe a class of three-dimensional theories which are realized as supersymmetric domain walls in the previously studied four-dimensional theories. This leads to an understanding of quantum field theories constructed from the six-dimensional (2; 0) superconformal field theory compactified on a three-manifold, and we develop the associated geometric dictionary. We find that the structure of the field theory is determined by a decomposition of the three-manifold into tetrahedra and a braid which species the relationship between ultraviolet and infrared geometries. The phenomenon of BPS wall-crossing in four dimensions is then seen in these domain walls to be responsible for three-dimensional mirror symmetries. / Physics