This thesis investigates the performance of cognitive radio relay networks (CRRN) in Rayleigh fading channel under various power constraints. Here spectrum sharing approach is considered, whereby a secondary user (SU) may be allowed to transmit simultaneously with a primary user (PU) as long as SU interference to PU remains below a tolerable level. In addition, SU has to meet certain quality of service (QoS) constraints of its own link. To support these QoS constraints, the maximal data rate that can be reliably transmitted with arbitrarily small error of probability is found. It is observed that this capacity is affected by channel quality and interference limit allowed by PU. Ergodic capacity and outage capacity which are two well known capacities, are analysed for CRRN under interference power constraints. This thesis also finds effective capacity for CRRN, a link layer channel model that models the effect of channel fading on queuing behaviour of the link. Effective capacity under interference and secondary transmitter power constraints is also investigated. The way of analysing effective capacity under interference and transmit power constraints is extended to ergodic capacity and outage capacity. Here it is observed that, capacity is affected by the minimum of transmit power and interference power constraints. Monte-Carlo simulations are carried out to support theoretical results obtained in this thesis.