A relay channel consists of an input , a relay output , a channel output , and a relay sender (whose transmission is allowed to depend on the past symbols . The dependence of the received symbols upon the inputs is given by . The channel is assumed to be memoryless. In this paper the following capacity theorems are proved. 1)If is a degraded form of , then . 2)If is a degraded form of , then . 3)If is an arbitrary relay channel with feedback from to both , then . 4)For a general relay channel, . Superposition block Markov encoding is used to show achievability of , and converses are established. The capacities of the Gaussian relay channel and certain discrete relay channels are evaluated. Finally, an achievable lower bound to the capacity of the general relay channel is established.