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In this paper, an reconfigurable intelligent surface (RIS)-aided millimeter wave (mmWave) non-orthogonal multiple access (NOMA) system is considered. In particular, we consider an RIS-aided mmWave-NOMA downlink system with a hybrid beamforming structure. To maximize the achievable sum-rate under a minimum rate constraint for the users and a minimum transmit power constraint, a joint RIS phase shifts, hybrid beamforming, and power allocation problem is formulated. To solve this non-convex optimization problem, we develop an alternating optimization algorithm. Specifically, first, the non-convex problem is transformed into three subproblems, i.e., power allocation, joint phase shifts and analog beamforming optimization, and digital beamforming design. Then, we solve the power allocation problem under fixed phase shifts of the RIS and hybrid beamforming. Finally, given the power allocation matrix, an alternating manifold optimization (AMO)-based method and a successive convex approximation (SCA)-based method are utilized to design the phase shifts, analog beamforming, and transmit beamforming, respectively. Numerical results reveal that the proposed alternating optimization algorithm outperforms state-of-the-art schemes in terms of sum-rate. Moreover, compared to a conventional mmWave-NOMA system without RIS, the proposed RIS-aided mmWave-NOMA system is capable of improving the achievable sum-rate of the system.