This thesis has four chapters. Chapter 1 introduces the concepts of Noetherianity up to the action of a group and polynomial functors. In Chapter 2, we prove that certain spaces associated to sequences of diagonal embeddings are Noetherian up to the action of the groups acting on them. Chapter 3 is devoted to investigating the strength of polynomials. In Chapter 4, we study closed subsets of polynomial functors. We prove that strict closed subsets always arise as unions of images from smaller polynomial functors.