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The Rayleigh flat fading channel at low SNR is considered. With full channel state information (CSI) at the transmitter and receiver, its capacity is shown to be essentially SNR log(1SNR) nats/symbol, as SNR goes to zero. In fact, this rate can be achieved with a just one bit of CSI at the transmitter (per fading realization) and with no receiver CSI. The capacity for the case of noisy transmitter CSI is also found. Then a Rayleigh block fading channel of coherence interval T ≤ 1/ SNR is considered which has causal feedback and no a priori CSI. A training based scheme is proposed for such channels, which achieves a rate of SNR logT nats/symbol in the limit of small SNR and large T. Thus, when coherence interval T is of the order 1/SNR, without any a priori CSI at either end, the capacity with full CSI at both ends is achievable. For smaller values of T, a rate of SNR logT nats/symbol is shown to be achievable.