Let f be an area-preserving surface diffeomorphism, and let Y be the mapping torus of f. We define a new version of symplectic Floer homology, which we call ”periodic Floer homology” (joint work with Michael Thaddeus). This is the homology of a chain complex in which the chains are generated by unions of periodic orbits of f, and the differential counts embedded pseudoholomorphic curves in R times Y. Our original motivation for studying this theory is that it is conjectured to agree with the Seiberg-Witten Floer homology of Y. Another motivation for studying this theory is that it has a formal analogue for a contact 3-manifold Y, which is a variant of the symplectic field theory of Eliashberg, Givental, and Hofer.