Overview
Representation theory looks at ways to study (usually hard) objects by “representing” them as linear transformations on a vector space. This is nice since we (pretty well) understand vector spaces and linear transformations, so we try to extend this knowledge as far as possible.
Representations in specific categories
There end up being some interesting nuances between representations in different categories, each with interesting points and emphases. So, instead of trying to stuff all the info in this page, here is a running list of types of representations that I have studied: