  We consider Sobolev spaces with values in Banach spaces as they are
frequently useful in applied problems. Given two Banach spaces $X\neq\{0\}$ and
$Y$, each Lipschitz continuous mapping $F:X\rightarrow Y$ gives rise to a
mapping $u\mapsto F\circ u$ from $W^{1,p}(\Omega,X)$ to $W^{1,p}(\Omega,Y)$ if
and only if $Y$ has the Radon-Nikodym Property. But if $F$ is one-sided Gateaux
differentiable no condition on the space is needed. We also study when weak
properties in the sense of duality imply strong properties. Our results are
applied to prove embedding theorems, a multi-dimensional version of the
Aubin-Lions Lemma and characterizations of the space $W^{1,p}_0(\Omega,X)$.
