We discuss the problem of designing channel access architectures for enabling fast, low-latency, grant-free and uncoordinated uplink for densely packed wireless nodes. Specifically, we study random-access codes, previously introduced for the AWGN multiple-access channel (MAC) by Polyanskiy'2017, in the practically more relevant case of users subject to Rayleigh fading, when channel gains are unknown to the decoder. We propose a random coding achievability bound, which we analyze both non-asymptotically (at finite blocklength) and asymptotically. As a candidate practical solution, we propose an explicit sparse-graph based coding scheme together with an alternating belief-propagation decoder. The latter's performance is found to be surprisingly close to the finite-blocklength bounds. Our main findings are twofold. First, just like in the AWGN MAC we see that jointly decoding large number of users leads to a surprising phase transition effect, where at spectral efficiencies below a critical threshold (5-15 bps/Hz depending on reliability) a perfect multi-user interference cancellation is possible. Second, while the presence of Rayleigh fading significantly increases the minimal required energy-per-bit $E_b/N_0$ (from about 0-2 dB to about 8-11 dB), the inherent randomization introduced by the channel makes it much easier to attain the optimal performance via iterative schemes. In all, it is hoped that a principled definition of the random-access model together with our information-theoretic analysis will open the road towards unified benchmarking and comparison performance of various random-access solutions, such as the currently discussed candidates (MUSA, SCMA, RSMA) for the 5G/6G.