  We explore single-particle Anderson localization due to nonrandom
quasiperiodic potentials in two and three dimensions. We introduce a class of
self-dual models that generalize the one-dimensional Aubry-Andr\'e model to
higher dimensions. In three dimensions (3D) we find that the Anderson
localization transitions appear to be in the same universality class as for
random potentials. In scaling or renormalization group terms, this means that
randomness of the potential is irrelevant at the Anderson localization
transitions in 3D. In two dimensions (2D) we also explore the Ando model, which
is in the symplectic symmetry class and shows an Anderson localization
transition for random potentials. Here, unlike in 3D, we find that the
universality class changes when we instead use a quasiperiodic potential.
