  In view of classical results of Masur and Veech almost every element in the
moduli space of compact translation surfaces is recurrent. In this paper we
focus on the problem of recurrence for elements of smooth curves in the moduli
space. We give an effective criterion for the recurrence of almost every
element of a smooth curve. The criterion relies on results developed by
Minsky-Weiss in \cite{Mi-We}. Next we apply the criterion to the billiard flow
on planar tables confined by arcs of confocal conics. The phase space of such
billiard flow splits into invariant subsets determined by caustics. We prove
that for almost every caustic the billiard flow restricted to the corresponding
invariant set is uniquely ergodic. This answers affirmatively to a question
raised by Zorich.
