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Cognitive radio networks hold the key to achieving better radio bandwidth utilization and improving the quality of wireless applications. The next step in this fast emerging paradigm is the multi-hop cognitive radio network. Well designed multi-hop cognitive radio networks can provide high bandwidth efficiency by using dynamic spectrum access technologies as well as provide extended coverage and ubiquitous connectivity for the wireless end users. However, the special features of multi-hop cognitive radio networks also raise several unique design challenges. In this article, we survey these unique challenges and open research issues in the design of multi-hop cognitive radio networks as well as discuss potential approaches to address these challenges. This article specifically focuses on the medium access control (MAC) and network layers of the multi-hop cognitive radio protocol stack. Issues considered include efficient spectrum sharing, optimal relay node selection, interference mitigation, end-to-end delay, etc.