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A network of three nodes mutually communicating with each other is studied. This multi-way network is a suitable model for three-user device-to-device communications. The main goal of this paper is to characterize the capacity region of the underlying Gaussian three-way channel (3WC) within a constant gap. To this end, a capacity outer bound is derived using cut-set bounds and genie-aided bounds. For achievability, the 3WC is first transformed into an equivalent star channel. This latter is then decomposed into a set of “successive” sub-channels, leading to a sub-channel allocation problem. Using backward decoding, interference neutralization, and known results on the capacity of the star-channel relying of physical-layer network coding, an achievable rate region for the 3WC is obtained. It is then shown that the achievable rate region is within a constant gap of the developed outer bound, leading to the desired capacity approximation. Interestingly, in contrast to the Gaussian two-way channel (TWC), adaptation is necessary in the 3WC. Furthermore, message splitting is another ingredient of the developed scheme for the 3WC, which is not required in the TWC. The two setups are, however, similar in terms of their sum-capacity pre-log, which is equal to 2. Finally, some interesting networks and their approximate capacities are recovered as special cases of the 3WC, such as the cooperative broadcast channel and multiple access channel. [ABSTRACT FROM PUBLISHER]