This dissertation studies the problem of wideband spectrum sensing for cognitive radio by partitioning into four fundamental elements: system modeling, performance metrics, sampling schemes, and detection algorithms. Each element can potentially couple individual channels, and appropriate designs of wideband spectrum sensing should consider the four elements jointly. We propose a p-sparse model to characterize the primary occupancy in a band of channels as a Bernoulli process, and suggest a pair of new performance metrics more appropriate for wideband spectrum sensing, specifically, the probability of insufficient spectrum opportunities PISO and the probability of excessive interference opportunities PEIO. We suggest two narrower band Nyquist sampling schemes with correspondingly much lower rates than wideband Nyquist rate, i.e., partial-band Nyquist sampling (PBNS) and sequential narrow band Nyquist sampling (SNNS), and establish a unified sub-Nyquist sampling structure, within which we study several important sub-Nyquist sampling schemes in literature. We investigate the aliasing patterns inherent in sub-Nyquist sampling and identify two extremes, specifically, uniform aliasing and periodic aliasing, and develop corresponding detection algorithms that allow tradeoffs between primary protection and secondary opportunities relevant to the goal of channel detection characterized Pm, the probability of missed detection, and Pf, the probability of false alarm, as well as the goal of wideband detection characterized by PISO and PEIO. For performance metrics that couple individual channels, multi-channel detection algorithms have an advantage over channel-by-channel detection algorithms even for Nyquist sampling that give independent observations across channels. Most importantly, integer undersampling (IU), which corresponds to the simplest sub-Nyquist sampling scheme, exhibits the best observed sensing performance in the regime of better protection for the primary system, i.e., the regime of low PM and high PF, or the regime of low PEIO and high PISO, for moderate and high signal-to-noise ratio (SNR &#8805; 0 dB); on the other hand, SNNS exhibits globally best performance for low SNR (< 0 dB) for the cases studied. These observations discourage studies on the design of more sophisticated sub-Nyquist sampling schemes and development of more advanced sparse reconstruction algorithms to the problem of wideband spectrum sensing, since their performance is inferior to either IU or SNNS depending on the system parameters and the detection regime considered.