Abstract: Measure equivalence is an equivalence relation on countable groups introduced by Gromov as a measure theoretic counterpart to the goemetric notion of quasi-isometry. In the first part of this talk I will give a brief introduction to measure equivalence. I will then discuss some new joint work with Lewis Bowen in which we show that the class B, of groups which satisfy the conclusion of Popa's Cocycle Superrigidity Theorem for Bernoulli shifts, is invariant under measure equivalence. As a consequence we show that any nonamenable lattice in a product of noncompact locally compact groups must belong to the class B. This also has implications for entropy: we introduce a new kind of entropy called weak Pinsker entropy, and show that equivalence relations generated by free measure preserving actions of groups in the class B completely "remember" the weak Pinsker entropy of the action.