A class ofM-ary adaptive communication receivers, both taught and untaught, for signals whose waveforms are unknown {\em a priori} is described. The optimalM-ary taught receiver for binary-valued signals and the binary symmetric channel is studied in detail. It is shown that a bank ofMadaptive digital correlators (or adaptive digital matched filters--ADMF's) represents a good approximation to the set ofMoptimal filters of that receiver. The performance of a taught ADMF is analyzed in terms of its output signal-to-noise ratio and the required number of training transmissions. This performance is compared with that of the optimal filter for the same signal. It is shown that both the ADMF and the optimal filter converge asymptotically to the digital matched filter for the signal. The general untaught optimal adaptive receiver requires a number of calculations that grows exponentially with time, and therefore is impractical to implement. A self-adapting untaught receiver. Which teaches itself on the basis of its own decisions, is shown to be practical to implement in special cases. The performance of such a receiver is analyzed. It is shown that a proper setting of threshold circuit for this receiver can speed up its rate of convergence.