  We show that a version of the desingularization theorem of Hironaka holds for
certain classes of infinitely differentiable functions (essentially, for
subrings that exclude flat functions and are closed under differentiation and
the solution of implicit equations). Examples are quasianalytic classes,
introduced by E. Borel a century ago and characterized by the Denjoy-Carleman
theorem. These classes have been poorly understood in dimension > 1. Resolution
of singularities can be used to obtain many new results; for example,
topological Noetherianity, Lojasiewicz inequalities, division properties.
