  We discuss a new approach to putting supersymmetric theories on the lattice.
The basic idea is to start from a {\it twisted} formulation of the underlying
supersymmetric theory in which the fermions are represented as grassmann valued
antisymmetric tensor fields. The original supersymmetry algebra is replaced by
a twisted algebra which contains a scalar nilpotent supercharge $Q$.
Furthermore the action of the theory can then be written as the $Q$-variation
of some function. The case of ${\cal N}=2$ super Yang-Mills theory in two
dimensions is discussed in some detail. We then present our proposal for
discretizing this theory and derive the resultant lattice action. The latter is
local, free of spectrum doubling, gauge invariant and preserves the scalar
supercharge invariance exactly. Some preliminary numerical results are then
presented. The approach can be naturally generalized to yield a lattice action
for ${\cal N}=4$ super Yang-Mills in four dimensions.
