  We develop a multigraded variant of Castelnuovo-Mumford regularity. Motivated
by toric geometry, we work with modules over a polynomial ring graded by a
finitely generated abelian group. As in the standard graded case, our
definition of multigraded regularity involves the vanishing of graded
components of local cohomology. We establish the key properties of regularity:
its connection with the minimal generators of a module and its behavior in
exact sequences. For an ideal sheaf on a simplicial toric variety X, we prove
that its multigraded regularity bounds the equations that cut out the
associated subvariety. We also provide a criterion for testing if an ample line
bundle on X gives a projectively normal embedding.
