The focus of this work is multi-user systems where the combined multi-user signal is passed through a known non-linearity, e.g., a downlink RF power amplifier in satellite communication systems. A successive interference annulment (SIA) multi-user detector for non-linear multi-user systems is proposed and analyzed in terms of the asymptotic behavior of the conditional probability of error at each stage of successive decoding. The design of the SIA detector takes into account the geometry of the overall signal constellation and a particular user's bit is decoded using a single inner product operation. The run-time complexity of the proposed SIA detector is linear in the number of users, though the design complexity is exponential in the number of users. For a linear multi-user system, the SIA detector reduces to a serial interference cancellation (SIC) detector that maximizes the asymptotic conditional multi-user efficiency at each stage of successive decoding. Simulation results indicate that the SIA detector consistently outperforms conventional detection schemes where the non-linearity is ignored.