We consider the problem of redistributing data on homogeneous and heterogeneous rings of processors. The problem arises in several applications, after each invoca-tion of a load-balancing mechanism (but we do not discuss the load-balancing mechanism itself). We provide algorithms that aim at optimizing the data redistribution, both for unidirectional and bidirectional rings. One major contribution of the paper is that we are able to prove the opti-mality of the proposed algorithms in all cases except that of a bidirectional heterogeneous ring, for which the problem remains open.