  We prove a monomialization theorem for mappings in general classes of
infinitely differentiable functions that are called quasianalytic. Examples
include Denjoy-Carleman classes, the class of $\cC^\infty$ functions definable
in a polynomially bounded $o$-minimal structure, as well as the classes of
real- or complex analytic functions, and algebraic functions over any field of
characteristic zero. The monomialization theorem asserts that a mapping in a
quasianalytic class can be transformed to a mapping whose components are
monomials with respect to suitable local coordinates, by sequences of simple
modifications of the source and target -- local blowings-up and power
substitutions in the real cases, in general, and local blowings-up alone in the
algebraic or analytic cases. Monomialization is a version of resolution of
singularities for a mapping. We show that it is not possible, in general, to
monomialize by global blowings-up, even in the real-analytic case.
