  In this paper we study asymptotic behavior of solutions of obstacle problems
for $p-$Laplacians as $p\to \infty.$ For the one-dimensional case and for the
radial case, we give an explicit expression of the limit. In the n-dimensional
case, we provide sufficient conditions to assure the uniform convergence of
whole family of the solutions of obstacle problems either for data $f$ that
change sign in $\Omega$ or for data $f$ (that do not change sign in $\Omega$)
possibly vanishing in a set of positive measure.
