  We consider the automorphism groups of various Lorentzian lattices over the
Eisenstein, Gaussian, and Hurwitz integers, and in some of them we find
reflection groups of finite index. These provide new finite-covolume reflection
groups acting on complex and quaternionic hyperbolic spaces. Specifically, we
provide groups acting on CH^n for all n<6 and n=7, and on HH^n for n=1,2,3 and
5. We compare our groups to those discovered by Deligne and Mostow and show
that our largest examples are new. For many of these Lorentzian lattices we
show that the entire symmetry group is generated by reflections, and obtain a
description of the group in terms of the combinatorics of a lower-dimensional
positive-definite lattice. The techniques needed for our lower-dimensional
examples are elementary, but to construct our best examples we also need
certain facts about the Leech lattice. We give a new and geometric proof of the
classifications of selfdual Eisenstein lattices of dimension < 7 and of
selfdual Hurwitz lattices of dimension < 5.
