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In this paper, we investigate covert communication in an intelligent reflecting surface (IRS)-assisted non-orthogonal multiple access (NOMA) system, where a legitimate transmitter (Alice) applies NOMA for downlink and uplink transmissions with a covert user (Bob) and a public user (Roy) aided by an IRS. Specifically, we propose new IRS-assisted downlink and uplink NOMA schemes to hide the existence of Bob's covert transmission from a warden (Willie), which cost-effectively exploit the phase-shift uncertainty of the IRS and the non-orthogonal signal transmission of Roy as the cover medium without requiring additional uncertainty sources. Assuming the worst-case covert communication scenario where Willie can optimally adjust the detection threshold for his detector, we derive an analytical expression for the minimum average detection error probability of Willie achieved by each of the proposed schemes. To further enhance the covert communication performance, we propose to maximize the covert rates of Bob by jointly optimizing the transmit power and the IRS reflect beamforming, subject to given requirements on the covertness against Willie and the quality-of-service (QoS) at Roy. Simulation results demonstrate the covertness advantage of the proposed schemes and confirm the accuracy of the derived analytical results. Interestingly, it is found that covert communication is impossible without using IRS or NOMA for the considered setup while the proposed schemes can always guarantee positive covert rates.