Trellis coding is investigated for a class of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> -user multiple-access channels. It is shown that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> -user trellis coding, in conjunction with Viterbi or sequential decoding, permits the multiple-access function to be combined with forward error correction. The distance and rate properties are analyzed for linear (convolutional) <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> -user trellis codes, and the use of nonlinear <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> -user trellis coding is discussed. A procedure is given for constructing <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> -user convolutional-code <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> -tuples with large free distance. Finally the error performance is analyzed, and coding gains of several decibels over uncoded time sharing are verified by simulation results.