We prove the following: there are infinitely many finite-covolume (resp. cocompact) Coxeter groups acting on hyperbolic space Hn for every n ≤ 19 (resp. n ≤ 6). When n = 7 or 8, they may be taken to be nonarithmetic. Furthermore, for 2 ≤ n ≤ 19, with the possible exceptions n = 16 and 17, the number of essentially distinct Coxeter groups in Hn with noncompact fundamental domain of volume ≤ V gr...