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Inspired by recent developments in full-duplex (FD) communications, we consider an opportunistic spectrum access (OSA) network in which secondary users (SUs) are capable of partial/complete self-interference suppression (SIS). This enables them to operate in either simultaneous transmit-and-sense (TS) or simultaneous transmit-and-receive (TR) modes, with the goal of achieving improved primary user (PU) detection and/or higher SU throughput. We first consider an overlay OSA setup, and we study the TS and TR modes. We also explore the spectrum awareness/efficiency tradeoff and determine an efficient adaptive strategy for the SU link. We then consider a spectrum underlay model, with the objective of optimizing SUs’ transmission powers so as to maximize the sum-throughput of K FD secondary links subject to a PU outage constraint. Operating in an FD fashion is not always efficient for SUs. Hence, we propose an optimal policy for switching between FD and half-duplex. The criteria for this policy depend mainly on the SIS capabilities of SUs. Finally, we propose a mode selection algorithm for the switching process. Numerical results indicate that operating in the TS mode can reduce the PU outage probability by up to 100% compared with the classical listen-before-talk scheme.