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Cognitive radio (CR) has emerged as a promising technology to exploit the unused portions of spectrum in an opportunistic manner. The fixed spectrum allocation of governmental agencies results in unused portions of spectrum, which are called "spectrum holes" or "white spaces". CR technology overcomes this issue, allowing devices to sense the spectrum for unused portions and use the most suitable ones, according to some pre-defined criteria. Spectrum assignment is a key mechanism that limits the interference between CR devices and licensed users, enabling a more efficient usage of the wireless spectrum. Interference is a key factor that limits the performance in wireless networks. The scope of this work is to give an overview of the problem of spectrum assignment in cognitive radio networks, presenting the state-of-the-art proposals that have appeared in the literature, analyzing the criteria for selecting the most suitable portion of the spectrum and showing the most common approaches and techniques used to solve the spectrum assignment problem. Finally, an analysis of the techniques and approaches is presented, discussing also the open issues for future research in this area.