With the rapid growth of demand for ever increasing data rates, spectrum resources have become more and more scarce. As a promising technique to increase the efficiency of the spectrum utilization, cognitive radio has great potential to meet such a requirement by allowing unlicensed users to coexist in licensed bands. In conventional CR systems, spectrum sensing is performed at the beginning of each time slot before data transmission. Unfortunately, this results in two major problems: transmission time reduction due to sensing, and sensing accuracy impairment due to data transmission. To tackle these problems, in this article we present a new design paradigm for future CR by exploring full duplex techniques to achieve simultaneous spectrum sensing and data transmission. With FD radios equipped at secondary users (SUs), the SUs can simultaneously sense and access the vacant spectrum, and thus significantly improve sensing performance while increasing data transmission efficiency. The aim of this article is to transform the promising conceptual framework into a practical wireless network design by addressing a diverse set of challenges such as protocol design and theoretical analysis. Several application scenarios with FD-enabled CR are elaborated, and key open research directions and novel algorithms in these systems are discussed.