Traditional wireless networks are regulated by a fixed spectrum assignment policy. This results in situations where most of the allocated radio spectrum is not utilized. In order to address this spectrum underutilization, cognitive radio (CR) has emerged as a promising solution. Spectrum sensing is an essential component in CR networks to discover spectrum opportunities. The most common spectrum sensing techniques are energy detection, matched filtering or cyclostationary feature detection, which aim to maximize the probability of detection subject to a certain false alarm rate. Besides probability of detection, detection delay is also a crucial criterion in spectrum sensing. In an interweave CR network, quick detection of the absence of primary user (PU), which is the owner of the licensed spectrum, allows good utilization of unused spectrum, while quick detection of PU transmission is important to avoid any harmful interference. This thesis consider quickest spectrum sensing, where the aim is to detect the PU with minimal detection delay subject to a certain false alarm rate. In the earlier chapters of this thesis, a single antenna cognitive user (CU) is considered and we study quickest spectrum sensing performance in Gaussian channel and classical fading channel models, including Rayleigh, Rician, Nakagami-m and a long-tailed channel. We prove that the power of the complex received signal is a sufficient statistic and derive the probability density function (pdf) of the received signal amplitude for all of the fading cases. The novel derivation of the pdfs of the amplitude of the received signal for the Rayleigh, Rician and Nakagami-m channels uses an approach which avoids numerical integration. We also consider the event of a mis-matched channel, where the cumulative sum (CUSUM) detector is designed for a specific channel, but a different channel is experienced. This scenario could occur in CR network as the channel may not be known and hence the CUSUM detector may be experiencing a different channel. Simulations results illustrate that the average detection delay depends greatly on the channel but very little on the nature of the detector. Hence, the simplest time-invariant detector can be employed with minimal performance loss. Theoretical expressions for the distribution of detection delay for the time-invariant CUSUM detector, with single antenna CU are developed. These are useful for a more detailed analysis of the quickest spectrum sensing performance. We present several techniques to approximate the distribution of detection delay, including deriving a novel closed-form expression for the detection delay distribution when the received signal experiences a Gaussian channel. We also derive novel approximations for the distribution of detection delay for the general case due to the absence of a general framework. Most of the techniques are general and can be applied to any independent and identically distributed (i.i.d) channel. Results show that different signal-to-noise ratio (SNR) and detection delay conditions require different methods in order to achieve good approximations of the detection delay distributions. The remarkably simple Brownian motion approach gives the best approximation for longer detection delays. In addition, results show that the type of fading channel has very little impact on long detection delays. In later chapters of this thesis, we employ multiple receive antennas at the CU. In particular, we study the performance of multi-antenna quickest spectrum sensing when the received signal experiences Gaussian, independent and correlated Rayleigh and Rician channels. The pdfs of the received signals required to form the CUSUM detector are derived for each of the scenarios. The extension into multiple antennas allows us to gain some insight into the reduction in detection delay that multiple antennas can provide. Results show that the sensing performance increases with an increasing Rician K-factor. In addition, channel correlation has little impact on the sensing performance at high SNR, whereas at low SNR, increasing correlation between channels improves the quickest spectrum sensing performance. We also consider mis-matched channel conditions and show that the quickest spectrum sensing performance at a particular correlation coefficient or Rician K-factor depends heavily on the true channel irrespective of the number of antennas at the CU and is relatively insensitive to the channel used to design the CUSUM detector. Hence, a simple multi-antenna time-invariant detector can be employed. Based on the results obtained in the earlier chapters, we derive theoretical expressions for the detection delay distribution when multiple receive antennas are employed at the CU. In particular, the approximation of the detection delay distribution is based on the Brownian motion approach.