  A variety of descent and major-index statistics have been defined for
symmetric groups, hyperoctahedral groups, and their generalizations. Typically
associated to pairs of such statistics is an Euler--Mahonian distribution, a
bivariate generating function identity encoding these statistics. We use
techniques from polyhedral geometry to establish new multivariate
generalizations for many of the known Euler--Mahonian distributions. The
original bivariate distributions are then straightforward specializations of
these multivariate identities. A consequence of these new techniques are
bijective proofs of the equivalence of the bivariate distributions for various
pairs of statistics.
