IntroductionUnder utilization of radio spectrum in traditional wireless communications systems [30], along with the increasing spectrum demand from emerging wireless applications, is driving the development of new spectrum allocation policies for wireless communications. These new spectrum allocation policies, which will allow unlicensed users (i.e., secondary users) to access the radio spectrum when it is not occupied by licensed users (i.e., primary users) will be exploited by the cognitive radio (CR) technology. Cognitive radio will improve spectrum utilization in wireless communications systems while accommodating the increasing amount of services and applications in wireless networks. A cognitive radio transceiver is able to adapt to the dynamic radio environment and the network parameters to maximize the utilization of the limited radio resources while providing flexibility in wireless access [45]. The key features of a CR transceiver include awareness of the radio environment (in terms of spectrum usage, power spectral density of transmitted/received signals, wireless protocol signaling) and intelligence. This intelligence is achieved through learning for adaptive tuning of system parameters such as transmit power, carrier frequency, and modulation strategy (at the physical layer), and higher-layer protocol parameters.Implementation of a cognitive radio will be based on the concept of dynamic spectrum access (DSA). Through DSA, frequency spectrum can be shared among primary users and cognitive radio users (i.e., secondary users) in a dynamically changing radio environment. There are two major flavors 2 E. Hossain, L. Le, N. Devroye, and M. Vu of dynamic spectrum access: dynamic licensing (for dynamic exclusive use of radio spectrum) and dynamic sharing (for coexistence) [3,120]. Dynamic sharing can be of two types: horizontal spectrum sharing and vertical spectrum sharing. In the former case, all users/nodes have equal regulatory status while in the latter case all users/nodes do not have equal regulatory status (i.e., there are primary users and secondary users) and secondary users opportunistically access the spectrum without negatively affecting the primary users' performance.In this chapter, we focus on vertical spectrum sharing in a cognitive radio network. In particular, we outline the recent information theoretic advances pertaining to the limits of such networks. Information theory provides an ideal framework as well as tools and metrics for analyzing the fundamental limits of communication. The limits obtained provide benchmarks for the operation of cognitive networks, allowing researchers and engineers to gauge the efficiency of any practical network and guide their design. Spectrum sensing is one of the major functions of a cognitive radio the goal of which is to determine the activity of licensed user by periodically observing signals on the target frequency bands. We discuss some theoretical results on the effect of side information (e.g., spatial locations of the users, transmission probabil...