  Let $J$ be a semisimple Lie group with all simple factors of real rank at
least two. Let $\Gamma<J$ be a lattice. We prove a very general local rigidity
result about actions of $J$ or $\Gamma$. This shows that almost all so-called
"standard actions" are locally rigid. As a special case, we see that any action
of $\Gamma$ by toral automorphisms is locally rigid. More generally, given a
manifold $M$ on which $\Gamma$ acts isometrically and a torus $\Ta^n$ on which
it acts by automorphisms, we show that the diagonal action on $\Ta^n{\times}M$
is locally rigid.
  This paper is the culmination of a series of papers and depends heavily on
our work in \cite{FM1,FM2}. The reader willing to accept the main results of
those papers as "black boxes" should be able to read the present paper without
referring to them.
