  In this paper we study minimal and constant mean curvature (cmc) periodic
surfaces in H^2 x R. More precisely, we consider quotients of H^2 x R by
discrete groups of isometries generated by horizontal hyperbolic translations f
and/or a vertical translation T. In the quotient by the Z^2 subgroup of the
isometry group generated by any f and T, we prove an Alexandrov-type theorem
for cmc surfaces, i.e., an analysis of compact embedded cmc surfaces in such a
quotient. The remainder of the paper is devoted to construct examples of
periodic minimal surfaces in H^2 x R.
