We report about a systematic study of entanglement entropy in relativistic quantum field theory. For the case of a 1+1-dimensional critical system, whose continuum limit is a conformal field theory with central charge c, we re-derive the result S_A~(c/3) log(l)  when A is a finite interval of length l in an infinite system, and extend it to many other cases: finite systems,finite temperatures, and when A consists of an arbitrary number of disjoint intervals. For such a system away from its critical point, when the correlation length xi is  large but finite, we  show that S_A~ A (c/6) log(xi), where  Ais the number of boundary points of A. These results are verified for a free massive field theory and for integrable lattice models. The free-field results are extended to higher dimensions, and used to motivate a scaling form  for the singular part of the entanglement entropy near a quantum phase transition.