A basic result in the theory of black holes is Hawking's theorem on black hole topology which asserts that for 3+1 AF stationary black hole spacetimes obeying the dominant energy condition, cross sections S of the event must be spherical. The proof is a beautiful variational argument showing that if S is not spherical then it can be deformed to an outer trapped surface in the domain of outer communications, which is forbidden by basic results. The conclusion also holds, by a similar argument, for outermost apparent horizons in black hole spacetimes that are not necessarily stationary. Since Gauss-Bonnet is used, the results are restricted to 3+1. In this talk we present a recent result with Rick Schoen which extends Hawking's results to arbitrarily high dimensions, by showing that outermost apparent horizons must be of positive Yamabe type, i.e., must carry metrics of positive scalar curvature. In the time symmetric case, this follows from the minimal surface methodology of Schoen and Yau in their treatment of manifolds of positive scalar curvature. The present result, however, does not impose any restrictions on the extrinsic curvature of space. While the Jang equation is not used, a neat technique appearing in Schoen-Yau II proves useful.