  We define motivic analogues of Igusa's local zeta functions. These functions
take their values in a Grothendieck group of Chow motives. They specialize to
p-adic Igusa local zeta functions and to the topological zeta functions we
introduced several years ago. We study their basic properties, such as
functional equations, and their relation with motivic nearby cycles. In
particular the Hodge spectrum of a singular point of a function may be
recovered from the Hodge realization of these zeta functions.
