In this doctoral dissertation, we address the artificial spectrum shortage problem and devise PHY/MAC layer schemes for secondary users (SUs) with cognitive radio capabilities to harness the underutilized spectrum resources in the spectrum bands assigned to existing primary users (PUs) non-intrusively. PUs are oblivious to the existence of SUs; and they can set a predefined protection requirement or impose a unit cost per interruption on SU access. We first present transmission strategies that maximize the SU utilization of the spectrum band when the PU is idle under the PU packet collision probability constraint. We show the optimality of a threshold-base spectrum access strategy for any semi-Markov PU traffic pattern assuming accurate sensing outcomes. The insight is that SUs should transmit with high probability when access opportunity is good. We characterize the interplay between the PU behavior (idle/busy time distributions) and secondary throughput performance. When the SU sensing is imperfect, we propose a modified threshold-based spectrum access scheme, and an adaptive scheme that enables distributed sharing among multiple SUs without knowing the PU traffic pattern. Simulation results show that both schemes approach the optimal SU throughput performance while satisfying the PU protection requirement. Next, we present a utility-maximization framework for general sensing-transmission structures. The proposed framework rewards SUs for successfully transmitted packets and penalizes for collisions caused to the PU. It leverages information including both sensing outcomes and ACK packets of the secondary receiver; and is able to tackle problems such as imperfect collision detection and sensing overhead. We then show that the optimal sensing-transmission structure is threshold-based. Last, we venture beyond the listen-before-talk approaches and present a novel cognitive access framework based on PU feedback channel. The proposed framework enables multiple SUs to assess their accumulated interruption to the primary receiver directly without explicit coordination from PUs. It also enables SUs to achieve optimal exploitation of spectrum opportunity in totally distributed ways without relying on common control channels. Within this framework, we present a discounted distributed power control algorithm and study its convergence and efficiency with/without synchronization among multiple SUs.