In this paper, we propose a cache-aided nonorthogonal multiple access (NOMA) scheme for spectrally efficient downlink transmission. The proposed scheme not only reaps the benefits associated with NOMA and caching, but also exploits the data cached at the users for interference cancellation. As a consequence, caching can help to reduce the residual interference power, making multiple decoding orders at the users feasible. The resulting flexibility in decoding can be exploited for improved NOMA detection. We characterize the achievable rate region of cache-aided NOMA and derive the Pareto optimal rate tuples forming the boundary of the rate region. Moreover, we optimize cache-aided NOMA for minimization of the time required for completing file delivery. The optimal decoding order and the optimal transmit power and rate allocation are derived as functions of the cache status, the file sizes, and the channel conditions. Simulation results confirm that, compared to several baseline schemes, the proposed cache-aided NOMA scheme significantly expands the achievable rate region and increases the sum rate for downlink transmission, which translates into substantially reduced file delivery times.