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Covert transmission is studied for an intelligent reflecting surface (IRS) aided communication system, where Alice aims to transmit messages to Bob without being detected by the warden Willie. Specifically, an IRS is used to increase the data rate at Bob under a covert constraint. For the considered model, when Alice is equipped with a single antenna, the transmission power at Alice and phase shifts at the IRS are jointly optimized to maximize the covert transmission rate with either instantaneous or partial channel state information (CSI) of Willie's link. In addition, when multiple antennas are deployed at Alice, we formulate a joint transmit beamforming and IRS phase shift optimization problem to maximize the covert transmission rate. One optimal algorithm and two low-complexity suboptimal algorithms are proposed to solve the problem. Furthermore, for the case of imperfect CSI of Willie's link, the optimization problem is reformulated by using the triangle and the Cauchy-Schwarz inequalities. The reformulated optimization problems are solved using an alterative algorithm, semidefinite relaxation (SDR) and Gaussian randomization techniques. Finally, simulations are performed to verify our analysis. The simulation results show that an IRS can degrade the covert transmission rate when Willie is closer to the IRS than Bob.