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Cognitive radio (CR) is the enabling technology for supporting dynamic spectrum access: the policy that addresses the spectrum scarcity problem that is encountered in many countries. Thus, CR is widely regarded as one of the most promising technologies for future wireless communications. To make radios and wireless networks truly cognitive, however, is by no means a simple task, and it requires collaborative effort from various research communities, including communications theory, networking engineering, signal processing, game theory, software-hardware joint design, and reconfigurable antenna and radio-frequency design. In this paper, we provide a systematic overview on CR networking and communications by looking at the key functions of the physical (PHY), medium access control (MAC), and network layers involved in a CR design and how these layers are crossly related. In particular, for the PHY layer, we will address signal processing techniques for spectrum sensing, cooperative spectrum sensing, and transceiver design for cognitive spectrum access. For the MAC layer, we review sensing scheduling schemes, sensing-access tradeoff design, spectrum-aware access MAC, and CR MAC protocols. In the network layer, cognitive radio network (CRN) tomography, spectrum-aware routing, and quality-of-service (QoS) control will be addressed. Emerging CRNs that are actively developed by various standardization committees and spectrum-sharing economics will also be reviewed. Finally, we point out several open questions and challenges that are related to the CRN design.