  In this paper we discuss the obstacle problem for the $p$-Laplace operator.
We prove optimal growth results for the solution. Of particular interest is the
point-wise regularity of the solution at free boundary points. The most
surprising result we prove is the one for the $p$-obstacle problem: Find the
smallest $u$ such that $$ \hbox{div} (|\nabla u|^{p-2}\nabla u) \leq 0, \qquad
u\geq \phi, \qquad \hbox{in } B_1, $$ with $\phi \in C^{1,1}(B_1)$ and given
boundary datum on $\partial B_1$. We prove that the solution is uniformly
$C^{1,1}$ at free boundary points. Similar results are obtained in the case of
an inhomogeneity belonging to $L^\infty$. When applied to the corresponding
parabolic problem, these results imply that any solution which is Lipschitz in
time is $C^{1,\frac{1}{p-1}}$ in the spatial variables.
