Let $N$ be any closed, Riemannian manifold. In this paper we prove that, for most locally symmetric, nonpositively curved Riemannian manifolds $M$, and for every continuous map $f:N\rightarrow M$, the map $f$ is homotopic to a smooth map with Jacobian bounded by a universal constant, depending (as it must) only on Ricci curvature bounds of $N$. From this we deduce an extension of Gromov's Volume Comparison Theorem for negatively curved manifolds to (most) nonpositively curved, locally symmetric manifolds.