I will discuss invariant subrings of group actions on finitely generated k-algebras for k algebraically closed. The plan is first to focus on the case for finite groups acting on polynomial rings, before generalizing the ideas to present a sketch of Hilbert's proof for the case where the group is linearly reductive and the ring is just finitely generated over k. The talk will feature a bit of representation theory, but I will try to define the notions necessary for the proof. Speaker(s): Saket Shah (University of Michigan, Ann Arbor)