This article investigates the downlink communications of intelligent reflecting surface (IRS) assisted non-orthogonal multiple access (NOMA) systems. To maximize the system throughput, we formulate a joint optimization problem over the channel assignment, decoding order of NOMA users, power allocation, and reflection coefficients. The formulated problem is proved to be NP-hard. To tackle this problem, a three-step novel resource allocation algorithm is proposed. Firstly, the channel assignment problem is solved by a many-to-one matching algorithm. Secondly, by considering the IRS reflection coefficients design, a low-complexity decoding order optimization algorithm is proposed. Thirdly, given a channel assignment and decoding order, a joint optimization algorithm is proposed for solving the joint power allocation and reflection coefficient design problem. Numerical results illustrate that: i) with the aid of IRS, the proposed IRS-NOMA system outperforms the conventional NOMA system without the IRS in terms of system throughput; ii) the proposed IRS-NOMA system achieves higher system throughput than the IRS assisted orthogonal multiple access (IRS-OMA) systems; iii) simulation results show that the performance gains of the IRS-NOMA and the IRS-OMA systems can be enhanced via carefully choosing the location of the IRS.