To support the rapidly increasing number of mobile users and mobile multimedia services, and the related demands for bandwidth, wireless communication technology is facing a potentially scarcity of radio spectrum resources. However, spectrum measurement campaigns have shown that the shortage of radio spectrum is due to inefficient usage and inflexible spectrum allocation policies. Thus, to be able to meet the requirements of bandwidth and spectrum utilization, spectrum underlay access, one of the techniques in cognitive radio networks (CRNs), has been proposed as a frontier solution to deal with this problem. In a spectrum underlay network, the secondary user (SU) is allowed to simultaneously access the licensed frequency band of the primary user (PU) as long as the interference caused by the SU to the PU is kept below a predefined threshold. By doing so, the spectrum utilization can be improved significantly. Moreover, the spectrum underlay network is not only considered as the least sophisticated in implementation, but also can operate in dense areas where the number of temporal spectrum holes is small. Inspired by the above discussion, this thesis provides a performance analysis of spectrum underlay networks which are subject to interference constraints. The thesis is divided into an introduction part and five parts based on peer-reviewed international research publications. The introduction part provides the reader with an overview and background on CRNs. The first part investigates the performance of secondary networks in terms of outage probability and ergodic capacity subject to the joint outage constraint of the PU and the peak transmit power constraint of the SU. The second part evaluates the performance of CRNs with a buffered relay. Subject to the timeout probability constraint of the PU and the peak transmit power constraint of the SU, system performance in terms of end-to-end throughput, end-to-end transmission time, and stable transmission condition for the relay buffer is studied. The third part analyzes a cognitive cooperative radio network under the peak interference power constraint of multiple PUs with best relay selection. The obtained results readily reveal insights into the impact of the number of PUs, channel mean powers of the communication and interference links on the system performance. The fourth part studies the delay performance of CRNs under the peak interference power constraint of multiple PUs for point-to-point and point-to-multipoint communications. A closedform expression for outage probability and an analytical expression for the average waiting time of packets are obtained for point-to-point communications. Moreover, the outage probability and successful transmission probability for packets in point-to-multipoint communications are presented. Finally, the fifth part presents work on the performance analysis of a spectrum underlay network for a general fading channel. A lower bound on the packet timeout probability and the average number of transmissions per packet are obtained for the secondary network.