  Given a vector space with an action of a semi-simple Lie algebra, we can try
to "categorify" this representation, which means finding a category where the
generators of the Lie algebra act by functors. Such categorical representations
arise naturally in geometric representation theory and in modular
representation theory of symmetric groups. A framework for studying categorical
representations was introduced by Rouquier and Khovanov-Lauda. Their
definitions are algebraic/diagrammatic, but are connected to the topology of
quiver varieties by the work of Rouquier and Varagnolo-Vasserot. In this paper,
we give a survey of the above circle of ideas.
