  This work examines the performance gain achieved by deploying an intelligent
reflecting surface (IRS) in covert communications. To this end, we formulate
the joint design of the transmit power and the IRS reflection coefficients by
taking into account the communication covertness for the cases with global
channel state information (CSI) and without a warden's instantaneous CSI. For
the case of global CSI, we first prove that perfect covertness is achievable
with the aid of the IRS even for a single-antenna transmitter, which is
impossible without an IRS. Then, we develop a penalty successive convex
approximation (PSCA) algorithm to tackle the design problem. Considering the
high complexity of the PSCA algorithm, we further propose a low-complexity
two-stage algorithm, where analytical expressions for the transmit power and
the IRS's reflection coefficients are derived. For the case without the
warden's instantaneous CSI, we first derive the covertness constraint
analytically facilitating the optimal phase shift design. Then, we consider
three hardware-related constraints on the IRS's reflection amplitudes and
determine their optimal designs together with the optimal transmit power. Our
examination shows that significant performance gain can be achieved by
deploying an IRS into covert communications.
