In these lecture proceedings, we describe some of the fundamental mathematical concepts that underlie supersymmetric string theory and field theory, and their role in describing and testing dualities. We provide a pedagogical introduction to topological and holomorphic twisting, descent, and higher algebraic structures. Our primary examples are worldsheet theories of topological strings, namely the A- and B-models, which we briefly review. These proceedings are based on lectures given by the second author at TASI 2021.