Consider a countable group G acting ergodically by measure preserving transformations on a probability space (X,m), and let R_G be the corresponding orbit equivalence relation on X. We show a new rigidity phenomenon in the context of orbit equivalence: there exist group actions such that the equivalence relation R_G on X determines the group G and the action (X,m,G) uniquely, up to finite groups. The natural action of SL(n,Z) on the n-torus R^n/Z^n, for n>2, is one of such examples. We also give a list of examples of countable equivalence relations of type II_1, which cannot be generated (mod 0) by a free action of any group. This gives a negative answer to a long standing problem of Feldman and Moore.