In this paper, using relaying broadcast channels (RBCs) as component channels for non-orthogonal multiple access (NOMA) is proposed to enhance the performance of NOMA in single-input single-output (SISO) cellular downlink systems. To analyze the performance of the proposed scheme, an achievable rate region of a RBC with compress-and-forward (CF) relaying is newly derived based on the recent work of noisy network coding (NNC). Based on the analysis of the achievable rate region of a RBC with decode-and-forward (DF) relaying, CF relaying, or CF relaying with dirty-paper coding (DPC) at the transmitter, the overall system performance of NOMA equipped with RBC component channels is investigated. It is shown that NOMA with RBC-DF yields marginal gain and NOMA with RBC-CF/DPC yields drastic gain over the simple NOMA based on broadcast component channels in a practical system setup. By going beyond simple broadcast channel (BC)/successive interference cancellation (SIC) to advanced multi-terminal encoding including DPC and CF/NNC, far larger gains can be obtained for NOMA.