  The main purpose of this paper is to investigate the behaviour of fractional
integral operators associated to a measure on a metric space satisfying just a
mild growth condition, namely that the measure of each ball is controlled by a
fixed power of its radius. This allows, in particular, non--doubling measures.
It turns out that this condition is enough to build up a theory that contains
the classical results based upon the Lebesgue measure on euclidean space and
their known extensions for doubling measures. We start by analyzing the images
of the Lebesgue spaces associated to the measure. The Lipschitz spaces, defined
in terms of the metric, play a basic role too. For a euclidean space equipped
with one of these measures, we also consider the so-called `` regular\rq\rq
$\bmo$ space introduced by X. Tolsa. We show that it contains the image of a
Lebesgue space in the appropriate limit case and also that the image of the
space `` regular\rq\rq $\bmo$ is contained in the adequate Lipschitz space.
