**Abstract:** This talk is about the use of symmetry in the study of modules and free resolutions in commutative algebra and algebraic geometry, and specifically how it clarifies, organizes, and rigidifies calculations, and how it enables us to find finiteness in situations where it a priori does not seem to exist. I will begin the talk with an example coming from classical invariant theory and determinantal ideals using just some basic notions from linear algebra. Then I will explain some of my own work which builds on this setting in several directions. Finally, I'll discuss a recent program on twisted commutative algebras, developed jointly with Andrew Snowden, which formalizes the synthesis of representation theory and commutative algebra and leads to new finiteness results in seemingly infinite settings.