We prove that the set of cyclic vectors for a von Neumann algebra in a Hilbert spaceH is aG set, which is empty or dense. We obtain some corollaries, for instance: if (A ₁,A ₂ ...) is a sequence of von Neumann algebras inH, and if eachA _n has a cyclic vector and a separating vector, then there exists a vector inH which is cyclic and separating for eachA _n. For algebras of local observables, we improve the known results connecting the infinite type of the algebras and the existence of cyclic and separating vectors.