  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. In particular, 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.
