Cognitive Radio (CR) has received tremendous research attention over the past decade, both in the academia and industry, as it is envisioned as a promising solution to the problem of spectrum scarcity. ACR is a device that senses the spectrum for occupancy by licensed users(also called as primary users), and transmits its data only when the spectrum is sensed to be available. For the efficient utilization of the spectrum while also guaranteeing adequate protection to the licensed user from harmful interference, the CR should be able to sense the spectrum for primary occupancy quickly as well as accurately. This makes Spectrum Sensing(SS) one of the where the goal is to test whether the primary user is inactive(the null or noise-only hypothesis), or not (the alternate or signal-present hypothesis). Computational simplicity, robustness to uncertainties in the knowledge of various noise, signal, and fading parameters, and ability to handle interference or other source of non-Gaussian noise are some of the desirable features of a SS unit in a CR. In many practical applications, CR devices can exploit known structure in the primary signal. IntheIEEE802.22CR standard, the primary signal is a wideband signal, but with a strong narrowband pilot component. In other applications, such as military communications, and blue tooth, the primary signal uses a Frequency Hopping (FH)transmission. These applications can significantly benefit from detection schemes that are tailored for detecting the corresponding primary signals. This thesis develops novel detection schemes and rigorous performance analysis of these primary signals in the presence of fading. For example, in the case of wideband primary signals with a strong narrowband pilot, this thesis answers the further question of whether to use the entire wideband for signal detection, or whether to filter out the pilot signal and use narrowband signal detection. The question is interesting because the fading characteristics of wideband and narrowband signals are fundamentally different. Due to this, it is not obvious which detection scheme will perform better in practical fading environments. At another end of the gamut of SS algorithms, when the CR has no knowledge of the structure or statistics of the primary signal, and when the noise variance is known, Energy Detection (ED) is known to be optimal for SS. However, the performance of the ED is not robust to uncertainties in the noise statistics or under different possible primary signal models. In this case, a natural way to pose the SS problem is as a Goodness-of-Fit Test (GoFT), where the idea is to either accept or reject the noise-only hypothesis. This thesis designs and studies the performance of GoFTs when the noise statistics can even be non-Gaussian, and with heavy tails. Also, the techniques are extended to the cooperative SS scenario where multiple CR nodes record observations using multiple antennas and perform decentralized detection. In this thesis, we study all the issues listed above by considering both single and multiple CR nodes, and evaluating their performance in terms of(a)probability of detection error,(b) sensing-throughput trade off, and(c)probability of rejecting the null-hypothesis. We propose various SS strategies, compare their performance against existing techniques, and discuss their relative advantages and performance tradeoffs. The main contributions of this thesis are as follows: The question of whether to use pilot-based narrowband sensing or wideband sensing is answered using a novel, analytically tractable metric proposed in this thesis called the error exponent with a confidence level. Under a Bayesian framework, obtaining closed form expressions for the optimal detection threshold is difficult. Near-optimal detection thresholds are obtained for most of the commonly encountered fading models. Foran FH primary, using the Fast Fourier Transform (FFT) Averaging Ratio(FAR) algorithm, the sensing-through put trade off are derived in closed form. A GoFT technique based on the statistics of the number of zero-crossings in the observations is proposed, which is robust to uncertainties in the noise statistics, and outperforms existing GoFT-based SS techniques. A multi-dimensional GoFT based on stochastic distances is studied, which pro¬vides better performance compared to some of the existing techniques. A special case, i.e., a test based on the Kullback-Leibler distance is shown to be robust to some uncertainties in the noise process. All of the theoretical results are validated using Monte Carlo simulations. In the case of FH-SS, an implementation of SS using the FAR algorithm on a commercially off-the ¬shelf platform is presented, and the performance recorded using the hardware is shown to corroborate well with the theoretical and simulation-based results. The results in this thesis thus provide a bouquet of SS algorithms that could be useful under different CRSS scenarios.