We study an equivalence relation - Measure Equivalence (ME) - between countable groups, which was introduced by Gromov. One of the motivations of ME is that all lattices in the same locally compact group are Measure Equivalent. Our main result is ME rigidity of higher rank lattices: any countable group which is ME to a lattice in a simple Lie group G of higher rank, is commensurable to a lattice in G. We also show that Kazhdan's property T is a ME invariant, and that the ME-class of Z is the class of all countable amenable groups.