Intelligent reflecting surface (IRS) is a new and disruptive technology to achieve spectrum- and energy-efficient as well as cost-efficient wireless networks. This paper considers an IRS-assisted downlink non-orthogonal-multiple-access (NOMA) system. To optimize the rate performance and ensure user fairness, we maximize the minimum decoding signal-to-interference-plus-noise-ratio (i.e., equivalently the rate) of all users, by jointly optimizing the (active) transmit beamforming at the base station (BS) and the phase shifts (i.e., passive beamforming) at the IRS. A combined-channel-strength based user ordering scheme is first proposed to decouple the user-ordering design and the joint beamforming design. Efficient algorithms are further proposed to solve the formulated non-convex problem for the cases of a single-antenna BS and a multi-antenna BS, respectively, by leveraging the block coordinated decent and semidefinite relaxation (SDR) techniques. For the single-antenna BS case, the optimal solution for the power allocation at the BS and the asymptotically optimal solution for the phase shifts at the IRS are obtained in closed forms. For the multi-antenna BS case, it is shown that the rank of the SDR solution to the transmit beamforming design is upper bounded by two. Also, the convergence proof and the complexity analysis are given for the proposed algorithms. 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. In addition, numerical results demonstrate that the rate degradation due to the IRS's finite phase resolution is slight, and good rate fairness among users can be always guaranteed.