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In this paper the Shannon capacity of generalized-gamma (GG) fading channels is studied, under different adaptive transmission techniques. The GG is a generic distribution, which is considered to be appropriate for modeling fading/shadowing phenomena. Hence, assuming single (i.e., no diversity), as well as, dual-branch selection diversity receivers, novel expressions for the capacity under three adaptive transmission policies, namely optimal power and rate adaptation (OPRA), channel inversion with fixed rate (CIFR) and truncated CIFR (TIFR), are obtained. The derived expressions are in closed form and generalize previously presented results. The proposed analysis is accompanied by numerical results, clearly demonstrating the usefulness of the theoretical approach.