  The topological complexity ${\sf TC}(X)$ is a homotopy invariant of a
topological space $X$, motivated by robotics, and providing a measure of the
navigational complexity of $X$. The topological complexity of a connected sum
of real projective planes, that is, a high genus nonorientable surface, is
known to be maximal. We use algebraic tools to show that the analogous result
holds for connected sums of higher dimensional real projective spaces.
