In this lecture series we will explore the use of symmetry in the study of cohomological invariants that appear naturally in commutative algebra. Symmetries occur frequently in nature and have important structural implications, which is why it is critical to identify and take advantage of them whenever possible. Grassmannians and flag varieties, spaces of matrices or of more general tensors, binary forms, or monomial ideals, are among the objects that we understand quite well, and our understanding relies in an essential way on our ability to exploit their symmetries. The goal of my lectures is to discuss key examples and highlight a variety of open questions.