A general technique is proposed for the derivation of upper bounds on channel capacity. The technique is based on a dual expression for channel capacity where the maximization (of mutual information) over distributions on the channel input alphabet is replaced with a minimization (of average relative entropy) over distributions on the channel output alphabet. Every choice of an output distribution --- even if not the channel image of some input distribution --- leads to an upper bound on mutual information. The proposed approach is used in order to study multi-antenna flat fading channels with memory where the realization of the fading process is unknown at the transmitter and unknown (or only partially known) at the receiver. It is demonstrated that, for high signal-to-noise ratio (SNR), the capacity of such channels typically grows only double-logarithmically in the SNR. This is in stark contrast to the case with perfect receiver side information where capacity grows logarithmically in the SNR. To better understand this phenomenon and the rates at which it occurs, we introduce the fading number as the second order term in the high SNR asymptotic expansion of capacity, and derive estimates on its value for various systems. It is suggested that at rates that are significantly higher than the fading number, communication becomes extremely power inefficient, thus posing a practical limit on the achievable rates. In an attempt to better understand the dependence of channel capacity on the fading law and on the number of antennae, we derive upper and lower bounds on the system's fading number. For Single-Input Single-Output (SISO) systems we present a complete characterization of the fading number for general stationary and ergodic fading processes. We also demonstrate that for memoryless Multi-Input Single-Output (MISO) channels, the fading number is achievable using beam-forming, and we derive an expression for the optimal beam direction. This direction depends on the fading law and is, in general, not the direction that maximizes the SNR on the induced SISO channel. Using a closed-form expression for the expectation of the logarithm of a non-central chi-square distributed random variable we provide some closed-form expressions for the fading number of some systems with Gaussian fading, including SISO systems with circularly symmetric stationary and ergodic Gaussian fading. The fading number of the latter is determined by the fading mean,fading variance, and the mean squared-error in predicting the present fading from its past; it is not directly related to the Doppler spread. A key ingredient in the analysis of the fading number is played by the notion of "capacity achieving input distributions that escape to infinity." This is a general property that many cost-constrained channel possess, and it is hoped that it will find use in the analysis of the high SNR behavior of other channels too. For some specific channels, e.g., the Rayleigh, Ricean, and Multi-Antenna Rayleigh fading channels we also present firm upper and lower bounds on channel capacity. These bounds are asymptotically tight in the sense that their difference from capacity approaches zero at high SNR, and their ratio to capacity approaches one at low SNR.