It is well known that the optimum receiver for pulse-amplitude-modulated (PAM) signals, transmitted over a single channel, consists of a cascade of continuous and transversal filters. It is shown that an extended structure of this type is also optimum for the completely general case ofMmultiplexed PAM signals transmitted over random-time and frequency-dispersive multichannel systems withIinputs andNoutputs, where each output is subjected to arbitrarily correlated additive noise. The random channels may also be arbitrarily correlated. In general, the receiver consists ofMNcontinuous filters andMtapped delay lines with M2sets of taps. An adaptive version is also described. The results include, as special cases, the situations considered by Gonsalves and Tufts , Shnidman , and Kaye . Also, included as special cases are new results for diversity systems, multi-input systems such as multipair cables, and multiple-terminal systems such as multiple access repeaters in which cochannel interference occurs.