This article contains a review of categorifications of semisimple
representations of various rings via abelian categories and exact endofunctors
on them. A simple definition of an abelian categorification is presented and
illustrated with several examples, including categorifications of various
representations of the symmetric group and its Hecke algebra via highest weight
categories of modules over the Lie algebra sl(n). The review is intended to
give non-experts in representation theory who are familiar with the topological
aspects of categorification (lifting quantum link invariants to homology
theories) an idea for the sort of categories that appear when link homology is
extended to tangles.Comment: latex, 35 pages, 4 eps figure