The possibility of parity-time (PT) symmetric periodic potentials is investigated within the context of optics. Beam dynamics in this new type of optical structures is examined in detail for both one- and two- dimensional lattice geometries. It is shown that PT periodic structures can exhibit unique characteristics stemming from the nonorthogonality of the associated Floquet-Bloch modes. Some of these features include double refraction, power oscillations, and eigenfunction unfolding as well as nonreciprocal diffraction patterns.