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Spectrum sensing aims at searching and finding the unused frequency bands in specific radio spectrum. It monitors the frequency bands to detect the activity of primary/licensed users and decide if secondary users can use these bands or not. In order to improve the efficiency of spectrum sensing in wideband cognitive radio networks, compressive sensing framework has been recommended and studied in many papers since it helps the system to get better and faster results using the sparse structure of the radio spectrum. Therefore, this paper represents an in-depth survey of the best requirements of compressive sensing and spectrum sensing techniques for robust combination and effective solution for wideband cognitive radio networks. It also provides examples of innovative applications of compressive spectrum sensing including IoT, smart city and 5th generation of mobile networks. To sum up some challenges …