  We study in this paper some local invariants attached via multiplier ideals
to an effective divisor or ideal sheaf on a smooth complex variety. First
considered (at least implicitly) by Libgober and by Loeser and Vaquie, these
jumping coefficients consist of an increasing sequence of rational numbers
beginning with the log canonical threshold of the divisor or ideal in question.
They encode interesting geometric and algebraic information, and we show that
they arise naturally in several different contexts. Given a polynomial f having
only isolated singularities, results of Varchenko, Loeser and Vaquie imply that
if \xi is a jumping number of f = 0 lying in the interval (0, 1], then -\xi is
a root of the Bernstein-Sato polynomial of f. We adapt an argument of Kollar to
show prove that this holds also when the singular locus of f has positive
dimension. In a more algebraic direction, we show that the number of such
jumping coefficients bounds the uniform Artin-Rees number of the principal
ideal (f) in the sense of Huneke: in the case of isolated singularities, this
in turn leads to bounds involving the Milnor and Tyurina numbers of f . Along
the way, we establish a general result relating multiplier to Jacobian ideals.
We also explore the extension of these ideas to the setting of graded families
of ideals. The paper contains many concrete examples.
