The time-reversal division multiple access (TRDMA) system employs multipath channels to realize multi-user transmission via time-reversal filtering. The performance of TRDMA is affected by the correlation among users’ channels. When the correlation is high, inter-user interference (IUI) in the TRDMA system will be serious and the system capacity will be severely restricted. In this paper, we propose an uplink multiple access scheme time-reversal non-orthogonal multiple access (TR-NOMA), wherein TRDMA and power-domain non-orthogonal multiple access are combined. The base station employs a TR filter for each user to suppress IUI and applies successive interference cancellations in the detection process of users’ signals. Users’ transmit power and TR filters are jointly optimized to promote system capacity. The original optimization problem is non-convex. We decompose the optimization problem into two sub-problems: the design of TR filters and the allocation of users’ transmit power, which is solved by utilizing generalized eigenvalue decomposition and successive lower bound convex approximation, respectively. The solution to the original optimization problem is obtained by the iterative solution between the two sub-problems. Simulation results demonstrate that the sum rate of the optimized TR-NOMA system is significantly higher than that of the TRDMA system.