It has been widely recognized that utilization of radio spectrum by licensed wireless systems, e.g., TV broadcasting, aeronautical telemetry, is quite low. In particular, at any given time and spatial region, there are frequency bands where there is no signal occupancy. There has been recent interest in improving spectrum utilization by permitting secondary usage using cognitive radios. Cognitive radios use spectrum sensing to determine frequency bands that are vacant of licensed signal transmissions and transmit on such portions to meet regulatory constraints of avoiding harmful interference to licensed systems. Future cognitive radios will be capable of scanning a wide band of frequencies, in the order of a few GHz, and employ adaptive waveforms for transmission depending on the estimated spectrum of licensed systems. In this thesis, we address the problem of estimating the spectrum of the wide-band signal received at the cognitive radio sensing receiver using compressive sampling coupled with a multiband spectrum detector to determine the spectrum occupancy of the licensed system. Since individual cognitive radios might not be able to reliably detect weak primary signals due to channel fading/shadowing, we also propose a distributed compressive scheme based on joint recovery of the license occupancy for application scenarios involving geographically distributed radios. In such a distributed approach, the spectrum occupancy is determined by the joint work of cognitive radios (exploiting spatial diversity), as opposed to being determined individually by each cognitive radio.