Two emerging technologies toward 5G wireless networks, namely non-orthogonal multiple access (NOMA) and cognitive radio (CR), will provide more efficient utilization of wireless spectrum in the future. In this article, we investigate the integration of NOMA with CR into a holistic system, namely a cognitive NOMA network, for more intelligent spectrum sharing. Design principles of cognitive NOMA networks are perfectly aligned to functionality requirements of 5G wireless networks, such as high spectrum efficiency, massive connectivity, low latency, and better fairness. Three different cognitive NOMA architectures are presented, including underlay NOMA networks, overlay NOMA networks, and CR-inspired NOMA networks. To address inter-network and intra-network interference, which largely degrade the performance of cognitive NOMA networks, cooperative relaying strategies are proposed. For each cognitive NOMA architecture, our proposed cooperative relaying strategy shows its potential to significantly lower outage probabilities. We discuss open challenges and future research directions on implementation of cognitive NOMA networks.