In this paper, nonorthogonal multiple access (NOMA) is applied to large-scale underlay cognitive radio (CR) networks with randomly deployed users. To characterize the performance of the considered network, new closed-form expressions of the outage probability are derived using stochastic geometry. More importantly, by carrying out the diversity analysis, new insights are obtained under the two scenarios with different power constraints: 1) fixed transmit power of the primary transmitters (PTs); and 2) transmit power of the PTs being proportional to that of the secondary base station. For the first scenario, a diversity order of m is experienced at the mth-ordered NOMA user. For the second scenario, there is an asymptotic error floor for the outage probability. Simulation results are provided to verify the accuracy of the derived results. A pivotal conclusion is reached that by carefully designing target data rates and power allocation coefficients of users, NOMA can outperform conventional orthogonal multiple access in underlay CR networks. © 2016 IEEE.