  We obtain a large class of cosmological solutions in the
toroidally-compactified low energy limits of string theories in $D$ dimensions.
We consider solutions where a $p$-dimensional subset of the spatial
coordinates, parameterising a flat space, a sphere, or an hyperboloid,
describes the spatial sections of the physically-observed universe. The
equations of motion reduce to Liouville or $SL(N+1,R)$ Toda equations, which
are exactly solvable. We study some of the cases in detail, and find that under
suitable conditions they can describe four-dimensional expanding universes. We
discuss also how the solutions in $D$ dimensions behave upon oxidation back to
the $D=10$ string theory or $D=11$ M-theory.
