  Using ideas borrowed from topological dynamics and ergodic theory we
introduce topological and metric versions of the recurrence property for
general Markov chains. The main question of interest here is how large is the
set of recurrent points. We show that under some mild technical assumptions the
set of non recurrent points is of zero reference measure. Necessary and
sufficient conditions for a reference measure $m$ (which needs not to be
dynamically invariant) to satisfy this property are obtained. These results are
new even in the purely deterministic setting.
