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The significant advances of cellular systems and mobile Internet services have yielded a variety of computation intensive applications, resulting in great challenge to mobile terminals (MTs) with limited computation resources. Mobile edge computing, which enables MTs to offload their computation tasks to edge servers located at cellular base stations (BSs), has provided a promising approach to address this challenging issue. Considering the advantage of improving transmission efficiency provided by nonorthogonal multiple access (NOMA), we propose an NOMA-enabled computation offloading scheme, in which a group of MTs offload partial of their computation workloads to an edge server based on the NOMA transmission. After finishing all MTs’ offloaded computation workloads, the edge server sends the computation results back to the MTs based on NOMA. We aim at minimizing the overall delay for completing all MTs’ computation requirements, which is achieved by jointly optimizing the MTs’ offloaded computation workloads, and the uploading duration for the MTs to send their computation workloads to the BS, and the downloading-duration for the BS to send the computation results back to the MTs. Despite the nonconvexity of the joint optimization problem, we exploit its layered structure and propose an efficient algorithm to compute the optimal offloading solution. Numerical results are provided to validate the accuracy and efficiency of our proposed algorithm and show the performance advantage of our NOMA-enabled computation-offloading scheme.