The emerging non-orthogonal multiple access (NOMA), which enables mobile users (MUs) to share same frequency channel simultaneously, has been considered as a spectrum-efficient multiple access scheme to accommodate tremendous traffic growth in future cellular networks. In this paper, we investigate the NOMA downlink relay-transmission, in which the macro base station (BS) first uses NOMA to transmit to a group of relays, and all relays then use NOMA to transmit their respectively received data to an MU. In specific, we propose an optimal power allocation problem for the BS and relays to maximize the overall throughput delivered to the MU. Despite the non-convexity of the problem, we adopt the vertical decomposition and propose a layered-algorithm to efficiently compute the optimal power allocation solution. Numerical results show that the proposed NOMA relay-transmission can increase the throughput up to 30 percent compared with the conventional time division multiple access (TDMA) scheme, and we find that increasing the relays’ power capacity can increase the throughput gain of the NOMA relay against the TDMA relay. Furthermore, to improve the throughput under weak channel power gains, we propose a hybrid NOMA (HB-NOMA) relay that adaptively exploits the benefit of NOMA relay and that of the interference-free TDMA relay. By using the throughput provided by the HB-NOMA relay for each individual MU, we study the multi-MUs scenario and investigate the multi-MUs scheduling problem over a long-term period to maximize the overall utility of all MUs. Numerical results demonstrate the performance advantage of the proposed multi-MUs scheduling that adopts the HB-NOMA relay-transmission.