  Given an effective Q-divisor D on a smooth complex variety, one can associate
to D its multiplier ideal sheaf J(D), which measures in a somewhat subtle way
the singularities of D. Because of their strong vanishing properties, these
ideals have come to play an increasingly important role in higher dimensional
geometry. We prove that for two effective Q-divisors D and E, one has the
"subadditivity" relation: J(D + E) \subseteq J(D) . J(E) . (We also establish
several natural variants, including the analogous statement for the analytic
multiplier ideals associated to plurisubharmonic functions.) As an application,
we give a new proof of a theorem of Fujita concerning the volume of a big
linear series on a projective variety. The first section of the paper contains
an overview of the construction and basic properties of multiplier ideals from
an algebro-geometric perspective, as well as a discussion of the relation
between some asymptotic algebraic constructions and their analytic
counterparts.
