During the last decade we have seen an explosive development of wireless technologies. Consequently the demand for electromagnetic spectrum has been growing dramatically resulting in the spectrum scarcity problem. In spite of this, spectrum utilization measurements have shown that licensed bands are vastly underutilized while unlicensed bands are too crowded. In this context, Cognitive Radio emerges as an auspicious paradigm in order to solve those problems. Even more, this concept is envisaged as one of the main components of future wireless technologies, such as the fifth-generation of mobile networks. In this regard, this thesis is founded on cognitive radio networks. We start considering a paid spectrum sharing approach where secondary users (SUs) pay to primary ones for the spectrum utilization. In particular, the first part of the thesis bears on the design and analysis of an optimal SU admission control policy, i.e. that maximizes the long-run profit of the primary service provider. We model the optimal revenue problem as a Markov Decision Process and we use dynamic programming (and other techniques such as sample-path analysis) to characterize properties of the optimal admission control policy. We introduce different changes to one of the best known dynamic programming algorithms incorporating the knowledge of the characterization. In particular, those proposals accelerate the rate of convergence of the algorithm when is applied in the considered context. We complement the analysis of the paid spectrum sharing approach using fluid approximations. That is to say, we obtain a description of the asymptotic behavior of the Markov process as the solution of an ordinary differential equation system. By means of the fluid approximation of the problem, we propose a methodology to estimate the optimal admission control boundary of the maximization profit problem mentioned before. In addition, we use the deterministic model in order to propose some tools and criteria that can be used to improve the mean spectrum utilization with the commitment of providing to secondary users certain quality of service levels. In wireless networks, a cognitive user can take advantage of either the time, the frequency, or the space. In the first part of the thesis we have been concentrated on timefrequency holes, in the second part we address the complete problem incorporating the space variable. In particular, we first introduce a probabilistic model based on a stochastic geometry approach. We focus our study in two of the main performance metrics: medium access probability and coverage probability. Finally, in the last part of the thesis we propose a novel methodology based on configuration models for random graphs. With our proposal, we show that it is possible to calculate an analytic approximation of the medium access probability (both for PUs and, most importantly, SUs) in an arbitrary large heterogeneous random network. This performance metric gives an idea of the possibilities offered by cognitive radio to improve the spectrum utilization. The introduced robust method, as well as all the results of the thesis, are evaluated by several simulations for different network topologies, including real scenarios of primary network deployments. Keywords: Markov decision process, fluid limit, stochastic geometry, random graphs,dynamic spectrum assignment, cognitive radio