  This expository article is devoted to the local theory of ultradifferentiable
classes of functions, with a special emphasis on the quasianalytic case.
Although quasianalytic classes are well-known in harmonic analysis since
several decades, their study from the viewpoint of differential analysis and
analytic geometry has begun much more recently and, to some extent, has earned
them a new interest. Therefore, we focus on contemporary questions closely
related to topics in local algebra. We study, in particular, Weierstrass
division problems and the role of hyperbolicity, together with properties of
ideals of quasianalytic germs. Incidentally, we also present a simplified proof
of Carleman's theorem on the non-surjectivity of the Borel map in the
quasianalytic case.
