Cognitive radio (CR) is the solution to the spectrum scarcity issue faced in wireless communications. It allows unlicensed secondary users (SUs) to opportunistically access the spectrum assigned to licensed primary users (PUs), but unused, on a non-interfering basis. Spectrum sensing is the process by which the CR node becomes aware of the spectrum occupancy, in order to decide which frequency bands to use. In the first part of this thesis, we consider the Bayesian changepoint detection theory to propose a new spectrum sensing algorithm that exploits the mobility of the SU and works for practical scenarios where the PU’s signal power is unknown to the SU. Simulation results show that the derived algorithm is a good choice when the SU could be required to detect very low signal-to-noise ratio (SNR) signals. In the second part, we introduce a new framework for joint transmission and sensing in mobile CR, by using changepoint detection for spectrum sensing. The optimal system parameters (sensing time, transmission time and detection threshold) that maximize the spectrum utilization are numerically computed and analyzed. In the last part, a new spectrum sensing algorithm, that exploits geolocation information about existing PUs by using the Bayesian block-based detection theory, is introduced. An approximate closed-form expression, based on Taylor series expansion, is provided. Numerical results show that the derived algorithm gives the minimum error probability when compared with an algorithm that does not use geolocation information.