Upper bounds on the capacity of flat fading channels are derived. The results apply to Rayleigh fading, to Ricean fading, and to more general circularly symmetric ergodic Gaussian fading. It is demonstrated that, contrary to the additive Gaussian noise channel where capacity grows logarithmically in the signal-to-noise ratio (SNR), the capacity of these channels typically grows only double-logarithmically in the SNR. Multi-antenna communication is also discussed, as also non-Gaussian fading.