In this paper, we propose bi-directional cooperative non-orthogonal multiple access (NOMA). Compared to conventional NOMA, the main contributions of bi-directional cooperative NOMA can be explained in two directions: 1) the proposed NOMA system is still efficient when the channel gains of scheduled users are almost the same and 2) the proposed NOMA system operates well without accurate channel-state information at the base station. In a two-user scenario, the closed-form ergodic capacity of hi-directional cooperative NOMA is derived, and it is proven to be better than those of other techniques. Based on the ergodic capacity, the algorithms to find optimal power allocations maximizing the user fairness and sum rate are presented. Outage probability is also derived, and we show that hi-directional cooperative NOMA achieves a power gain over uni-directional cooperative NOMA and a diversity gain over non-cooperative NOMA and orthogonal multiple access (OMA). We finally extend the bidirectional cooperative NOMA to a multiuser model. The analysis of ergodic capacity and outage probability in a two-user scenario is numerically verified. Also, simulation results show that bidirectional cooperative NOMA provides better data rates than the existing NOMA schemes as well as OMA in a multiuser scenario.