Discretization of supersymmetric theories is an old problem in lattice field
theory. It has resisted solution until quite recently when new ideas drawn from
orbifold constructions and topological field theory have been brought to bear
on the question. The result has been the creation of a new class of lattice
gauge theory in which the lattice action is invariant under one or more
supersymmetries. The resultant theories are local and free of doublers and in
the case of Yang-Mills theories also possess exact gauge invariance. In
principle they form the basis for a truly non-perturbative definition of the
continuum supersymmetric field theory. In this talk these ideas are reviewed
with particular emphasis being placed on N=4 super Yang-Mills theory.Comment: Plenary talk at the symposium Quantum Theory and Symmetries,
Lexington, Kentucky, July 2009. References adde
