In this paper, we propose a novel non-orthogonal multiple access (NOMA) scheme based on trellis-coded modulation (TCM). Different from those in the traditional code-domain NOMA, the incoming data streams of multiple users are jointly coded and mapped to the codewords so as to improve the coding gain of the system without any bandwidth extension. Both codeword design and multi-user detection are considered to optimize the TCM for NOMA. The multi-dimensional (MD) constellation is carefully selected and a new set partitioning algorithm is developed for MD signal set labeling. To achieve the trade-off between the BER performance and complexity, a suboptimal two-layer Viterbi algorithm is proposed for joint decoding. Simulation results show that our proposed TCM-based NOMA scheme performs significantly better than the traditional code-domain NOMA and OFDMA schemes in terms of the BER performance under the same channel conditions.