  Multi-time wave functions are wave functions for multi-particle quantum
systems that involve several time variables (one per particle). In this paper
we contrast them with solutions of wave equations on a space-time with multiple
timelike dimensions, i.e., on a pseudo-Riemannian manifold whose metric has
signature such as ${+}{+}{-}{-}$ or ${+}{+}{-}{-}{-}{-}{-}{-}$, instead of
${+}{-}{-}{-}$. Despite the superficial similarity, the two behave very
differently: Whereas wave equations in multiple timelike dimensions are
typically mathematically ill-posed and presumably unphysical, relevant
Schr\"odinger equations for multi-time wave functions possess for every initial
datum a unique solution on the spacelike configurations and form a natural
covariant representation of quantum states.
