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A discrete memoryless channel with two inputs and two outputs, called a two-user channel, is studied under the communication situation where only separate messages are allowed to be sent between two source-user pairs. An outer bound to the capacity region is obtained by a method similar to that used by the author for the broadcast channel. Two extreme cases of two-user channels are discussed: separate channels and incompatible channels. Degraded two-user channels are introduced and studied in detail; in particular, an achievable region is obtained by combining two regions that correspond to the two different modes of transmission. This idea is extended to the general two-user channel where an achievable region for the general channel is obtained by random coding arguments.