  The r-parallel set to a set A in a Euclidean space consists of all points
with distance at most r from A. We clarify the relation between the volume and
the surface area of parallel sets and study the asymptotic behaviour of both
quantities as r tends to 0. We show, for instance, that in general, the
existence of a (suitably rescaled) limit of the surface area implies the
existence of the corresponding limit for the volume, known as the Minkowski
content. A full characterisation is obtained for the case of self-similar
fractal sets. Applications to stationary random sets are discussed as well, in
particular, to the trajectory of the Brownian motion.
