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In this study, the authors consider the α − η − μ fading channels and derive a new form for the cumulative distribution function of the signal‐to‐noise ratio (SNR) in terms of the bivariate Fox H‐function. They also derive closed‐form expressions for the moments and the amount of fading of the SNR. The capacity of α − η − μ fading channels is analysed assuming different adaptive transmission protocols. To this end, they provide analytical closed‐form expressions for the average channel capacity with optimal rate adaptation (C_ORA), optimal power and rate adaptation (C_OPRA), channel inversion with fixed rate (CIFR) (C_CIFR), and truncated CIFR (C_TIFR). The derived expressions are valid for arbitrary values of the α, η, and μ fading parameters. The derived expressions can also be reduced to study the capacity of other fading distributions such as η − μ, α − μ, Weibull, and Nakagami. Numerical results …