  Group algebras of permutations have proved highly useful in solving a number
of problems in large N gauge theories. I review the use of permutations in
classifying gauge invariants in one-matrix and multi-matrix models and
computing their correlators. These methods are also applicable to tensor models
and have revealed a link between tensor models and the counting of branched
covers. The key idea is to parametrize $U(N)$ gauge invariants using
permutations, subject to equivalences. Correlators are related to group
theoretic properties of these equivalence classes. Fourier transformation on
symmetric groups by means of representation theory offers nice bases of
functions on these equivalence classes. This has applications in AdS/CFT in
identifying CFT duals of giant gravitons and their perturbations. It has also
lead to general results on quiver gauge theory correlators, uncovering links to
two dimensional topological field theory and the combinatorics of trace
monoids.
