  We associate a family of ideal sheaves to any Q-effective divisor on a
complex manifold, called the higher multiplier ideals, using the theory of
mixed Hodge modules and V-filtrations. This family is indexed by two
parameters, an integer indicating the Hodge level and a rational number, and
these ideals admit a weight filtration. When the Hodge level is zero, they
recover the usual multiplier ideals. We study the local and global properties
of higher multiplier ideals systematically. In particular, we prove vanishing
theorems and restriction theorems, and provide criteria for the nontriviality
of the new ideals. The main idea is to exploit the global structure of the
V-filtration along an effective divisor using the notion of twisted Hodge
modules. In the local theory, we introduce the notion of the center of minimal
exponent, which generalizes the notion of minimal log canonical center. As
applications, we prove some cases of conjectures by Debarre, Casalaina-Martin
and Grushevsky on singularities of theta divisors on principally polarized
abelian varieties and the geometric Riemann-Schottky problem.
