I review recent approaches to constructing supersymmetric lattice theories
focusing in particular on the concept of topological twisting. The latter
technique is shown to expose a nilpotent, scalar supersymmetry which can be
implemented exactly in the lattice theory. Using these ideas a lattice action
for N=4 super Yang-Mills in four dimensions can be written down
which is gauge invariant, free of fermion doublers and respects one out of a
total of 16 continuum supersymmetries. It is shown how these exact symmetries
together with the large point group symmetry of the lattice strongly constrain
the possible counterterms needed to renormalize the theory and hence determine
how much residual fine tuning will be needed to restore all supersymmetries in
the continuum limit. We report on progress to study these renormalization
effects at one loop. We go on to give examples of applications of these
supersymmetric lattice theories to explore the connections between gauge
theories and gravity.Comment: 16 pages. Plenary talk at Lattice 201
