Full duplex (FD) systems are able to transmit and receive signals over the same frequency band simultaneously with the potential of even doubling the spectral efficiency in comparison with traditional half duplex (HD) systems. When combined with relaying, FD systems are expected to dramatically increase the throughput of future wireless networks. However, the degrading effect of self-interference (SI) due to the simultaneous transmission and reception at the relay threatens their efficient rollout in real-world topologies. At the same time, non-orthogonal multiple access (NOMA) can further increase the spectral efficiency of the network. In this article, we present how buffering at the relays can be integrated with FD and NOMA in order to improve the performance of relay networks. More specifically, the key points for the successful integration of buffer-aided relays with spatial sharing paradigms are presented, and details on buffer-state-information-based relay selection and its fully distributed implementation are discussed. Furthermore, a mathematical framework for the analysis of FD and HD buffer-aided relay networks is rigorously discussed. Performance evaluation shows the importance of data buffering toward mitigating SI and improving the sum-rate of the network. Finally, the goal of this work is both to summarize the current state of the art and, by calling attention to open problems, to spark interest toward targeting these and related problems in relay networks.