  The nature and origin of exceptional sets associated with the rotation number
of circle maps, Kolmogorov-Arnol'd-Moser theory on the existence of invariant
tori and the linearisation of complex diffeomorphisms are explained. The
metrical properties of these exceptional sets are closely related to
fundamental results in the metrical theory of Diophantine approximation. The
counterpart of Diophantine approximation in hyperbolic space and a dynamical
interpretation which led to the very general notion of `shrinking targets' are
sketched and the recent use of flows in homogeneous spaces of lattices in the
proof of the Baker-Sprindzuk conjecture is described briefly.
