Due to the scarcity of frequency spectrum in view of the evolution of wireless communication technologies, the cognitive radio (CR) concept has been introduced to efficiently exploit the available spectrum. This concept consists in introducing unlicensed/secondary users (SU’s) in existing networks to share the spectrum of licensed/primary users (PU’s) without harming primary communications hence the name of “spectrum sharing” technique. We study in this dissertation, the capacity and the achievable rate of the secondary user within various communication settings. We, firstly, investigate the capacity of the (SU’s) at low power regime for Nakagami fading channels and present closed form of the capacity under various types of interference and/or power constraints. We explicitly characterize two regimes where either the interference constraint or the power constraint dictates the optimal power profile. Our framework also highlights the effects of different fading parameters on the secondary link ergodic capacity. Interestingly, we show that the low power regime analysis provides a specific insight on the capacity behavior of CR that has not been reported by previous studies. Next, we determine the spectral efficiency gain of an uplink CR Multi-Input Multi- Output (MIMO) system in which the SU is allowed to share the spectrum with the PU using a specific precoding scheme to communicate with a common receiver. Applied to Rayleigh fading channels, we show, through numerical results, that our proposed scheme enhances considerably the cognitive achievable rate. For instance, in case of a perfect detection of the PU signal, after applying Successive Interference Cancellation (SIC), the CR rate remains non-zero for high Signal to Noise Ratio (SNR) which is usually impossible when we only use space alignment technique. In addition, we show that the rate gain is proportional to the allowed interference threshold by providing a fixed rate even in the high SNR range. Finally, we study the impact of the broadcast approach and multi-layer coding on the throughput of CR systems for general fading channels. And we found that at the absence of the channel state information(CSI), we show that this improvement could be almost reached by 2-Layers coding. Then, we introduce a quantized CSI policy and highlight its improvement in terms of throughput before we study the rate when BA with quantized CSI is adopted. Numerical results show that the improvement of the additional layers is decreasing as the number of quantized regions increases.