DescriptionThis dissertation is devoted to the study of categorical aspects of BPS states in two-dimensional quantum field theories with N=(2,2) supersymmetry. The basic aim of a categorical discussion is to study spaces of BPS states, which carry much more refined information than their traditionally studied characters. In a two-dimensional theory, whereas BPS states are supersymmetric states defined on a one-dimensional spatial slice, carrying out the discussion at a categorical level requires one to incorporate two-dimensional supersymmetric instantons. We motivate these instantons and the differential equations they obey in a broader physical context. We show how these instanton effects can be incorporated to result in a categorification of the Cecotti-Vafa wall-crossing formula. We generalize the discussion to incorporate two-dimensional theories with non-trivial twisted mass terms. The presence of twisted masses require us to incorporate Fock spaces of periodic solitons into the discussion, and we show how these Fock spaces affect the categorical wall-crossing formalism. We sketch two important future directions. The first involves the application of the ideas of this thesis to the study of three-manifolds and homological knot invariants. The second has us graduate from two-dimensional theories and enter the world of four-dimensional N=2 theories and their BPS states.