  A pulse of light, injected into a weakly disordered dielectric medium,
typically, will leave its initial location in a short time, by diffusion.
However, due to some rare configurations of disorder, there is a possibility of
formation of high quality resonators which can trap light for a long time. We
present a rather detailed, quantitative study of such random resonators and of
the "almost localized" states that they can support. After presenting a brief
review of the earlier work on the subject, we concentrate on a detailed
computation of the "prefactor": knowledge of the latter is crucial for
varifying the viability of the random rasonators and their areal density. Both
short range disorder (white noise) and correlated disorder are studied, and the
important effect of the correlation radius, $R_c$, on the probability of
formation of resonators with a given quality factor $Q$ is discussed. The
random resonators are "self-formed", in the sense that no sharp features (like
Mie scatterers or other "resonant entities") are introduced: our model is a
featureless dielectric medium with fluctuating dielectric constant. We point
out the relevance of the random resonators to the recently discovered
phenomenon of coherent "random" lasing and review the existing work on that
subject. We emphasize, however, that the random resonators exist already in the
{\em passive} medium: gain is only needed to "make them visible".
