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. 2) The proposed NOMA system operates well without accurate channel state information (CSI) at the base station (BS). In a two-user scenario, the closed-form ergodic capacity of bi-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 user fairness and sum-rate are presented. Outage probability is also derived, and we show that bi-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 bi-directional cooperative NOMA to a multi-user model. The analysis of ergodic capacity and outage probability in two-user scenario is numerically verified. Also, simulation results show that bi-directional cooperative NOMA provdes better data rates than the existing NOMA schemes as well as OMA in multi-user scenario.