The recently introduced unlimited sampling theorem proves that a one-dimensional bandlimited function can be perfectly recovered from a constant factor oversampling of its modulo samples. The advantage of this approach is that arbitrary high-dynamic-range signals can be recovered without sensor saturation or clipping. In this paper, we prove a multidimensional version of the unlimited sampling theorem that works with arbitrary sampling lattices. We also present a geometrical perspective on the emerging class of modulo sampling problem that is based on the topology of quotient spaces.