  We survey ultradifferentiable extension theorems, i.e., quantitative versions
of Whitney's classical extension theorem, with special emphasis on the
existence of continuous linear extension operators. The focus is on
Denjoy-Carleman classes for which we develop the theory from scratch and
discuss important related concepts such as (non-)quasianalyticity. It allows us
to give an efficient and, to a fair extent, elementary introduction to
Braun-Meise-Taylor classes based on their representation as intersections and
unions of Denjoy-Carleman classes.
