  The main result is a classification of smooth actions of $SL(n,{\bf R})$, $n
\geq 3$, or connected groups locally isomorphic to it, on closed $n$-manifolds,
extending a theorem of Uchida. We construct new exotic actions of $SL(n,{\bf
Z})$ on the $n$-torus and connected sums of $n$-tori, and we formulate a
conjectural classification of actions of lattices in $SL(n,{\bf R})$ on closed
$n$-manifolds. We prove some results about invariant rigid geometric structures
for $SL(n,{\bf R})$-actions.

Differential Geometry