We describe a new approach to the problem of putting supersymmetric theories
on the lattice. The basic idea is to discretize a {\it twisted} formulation of
the supersymmetric theory. For certain theories with extended supersymmetry
these twisted formulations contain only integer spin fields. The twisting
exposes a scalar nilpotent supercharge which generates an exact lattice
symmetry. We gives examples from quantum mechanics, sigma models and Yang-Mills
theories.Comment: Summary of lectures given at the Summer Institute 2005 Fuji-Yoshida,
Japan August 11-18, 200
