Intelligent reflecting surface (IRS) is a new and disruptive technology to achieve spectrum-, energy, and cost-efficient wireless networks. In this paper, we consider an IRS-assisted non-orthogonal-multiple-access (NOMA) system in which a base station (BS) transmits superposed downlink signals to multiple users. A combined-channel-strength (CCS) based user ordering scheme is first proposed. In order to optimize the rate performance and ensure user fairness, we further maximize the minimum decoding signal-to-interference-plus-noise-ratio (i.e., equivalently the rate) of all users, by jointly optimizing the power allocation at the BS and the phase shifts at the IRS. However, the formulated problem is non-convex and difficult to be solved optimally. By leveraging the block coordinate descent and semidefinite relaxation techniques, an efficient algorithm is then proposed to obtain a suboptimal solution. Simulation results show that the IRS-assisted downlink NOMA system can enhance the rate performance significantly, compared to traditional NOMA without IRS and traditional orthogonal multiple access with/without IRS, and the rate degradation due to the IRS’s finite phase resolution is slight.