The explosive data traffic and connections in 5G networks require the use of non-orthogonal multiple access (NOMA) to accommodate more users. Unmanned aerial vehicle (UAV) can be exploited with NOMA to improve the situation further. In this paper, we propose a UAV-assisted NOMA network, in which the UAV and base station (BS) cooperate with each other to serve ground users simultaneously. The sum rate is maximized by jointly optimizing the UAV trajectory and the NOMA precoding. To solve the optimization, we decompose it into two steps. First, the sum rate of the UAV-served users is maximized via alternate user scheduling and UAV trajectory, with its interference to the BS-served users below a threshold. Then, the optimal NOMA precoding vectors are obtained using two schemes with different constraints. The first scheme intends to cancel the interference from the BS to the UAV-served user, while the second one restricts the interference to a given threshold. In both schemes, non-convex optimization problems are converted into tractable ones. An iterative algorithm is designed. Numerical results are provided to evaluate the effectiveness of the proposed algorithms for the hybrid NOMA and UAV network.