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Spectrum sensing is the key enabling technology for cognitive radio networks. The main objective of spectrum sensing is to provide more spectrum access opportunities to cognitive radio users without interfering with the operations of the licensed network. Hence, recent research has been focused on the interference avoidance problem. Moreover, current radio frequency (RF) front-ends cannot perform sensing and transmission at the same time, which inevitably decreases their transmission opportunities, leading to the so-called sensing efficiency problem. In this paper, in order to solve both the interference avoidance and the spectrum efficiency problem, an optimal spectrum sensing framework is developed. More specifically, first a theoretical framework is developed to optimize the sensing parameters in such a way as to maximize the sensing efficiency subject to interference avoidance constraints. Second, in order to exploit multiple spectrum bands, spectrum selection and scheduling methods are proposed where the best spectrum bands for sensing are selected to maximize the sensing capacity. Finally, an adaptive and cooperative spectrum sensing method is proposed where the sensing parameters are optimized adaptively to the number of cooperating users. Simulation results show that the proposed sensing framework can achieve maximum sensing efficiency and opportunities in multi-user/multi-spectrum environments, satisfying interference constraints.