  In dimensions $n\ge 2$ we obtain $L^{p_1}(\mathbb R^n) \times\dots\times
L^{p_m}(\mathbb R^n)$ to $L^p(\mathbb R^n)$ boundedness for the multilinear
spherical maximal function in the largest possible open set of indices and we
provide counterexamples that indicate the optimality of our results. Moreover,
we obtain weak type and Lorentz space estimates as well as counterexamples in
the endpoint cases.
