Wireless networking is fast becoming the primary method for people to connect to the Internet and with each other. The available wireless spectrum is increasingly loaded, with users demanding higher performance and reliability from their wireless connections. This dissertation proposes single channel full-duplex as a new paradigm for wireless system design that can mitigate some of the throughput and reliability problems of today's wireless systems. With full-duplex wireless radios, a node can receive and transmit data at the same time, without using multiple wireless channels. At the physical layer, this capability can double the available throughput at a node. Further, sending and receiving at the same time allows a wireless node to exchange control messages while receiving data, making real-time feedback schemes possible. Such feedback, assumed to be impossible till now, allows network researchers to rethink the way wireless networks are designed. This dissertation shows that with the extra feedback channel available through full-duplexing, reliability in existing wireless networks can be significantly improved. Motivated by the promise that full-duplex wireless holds, this dissertation explores the challenges in implementing such radios and how these radios could influence the design of wireless networking stacks. The primary challenge in implementing a full-duplex wireless system is removing the self-interference from a node's transmit antenna to its receive antenna. This dissertation discusses various analog and digital techniques to cancel this self-interference. It presents an adaptive full-duplex wireless system that combines analog and digital self-interference cancellation to remove up to 73dB of self-interference. Exploiting full-duplexing to the fullest extent requires a redesign of higher layers, especially the MAC and network layers. This dissertation presents a full-duplex MAC layer implementation to show how full-duplexing can be leveraged to improve wireless reliability and performance. This implementation reduces hidden terminal losses by up to 88% and significantly improves network fairness and throughput. The main contribution of this dissertation is to motivate full-duplex radios as a direction for research on future wireless systems. It shows that designing full-duplex systems, while challenging, is feasible. It also presents many ideas on how full-duplexing could be used to improve a variety of wireless systems, including multi-hop data networks, cellular systems, and wireless LANs.