We analyze the outage and throughput performance of full-duplex relay selection (FDRS) in underlay cognitive networks. Contrary to half-duplex relaying, full-duplex relaying (FDR) enables simultaneous listening/forwarding at the secondary relay(s), thereby allowing for higher spectral efficiency. However, due to simultaneous source/relay transmissions in FDR, the superimposed signal at the primary receiver should now satisfy the existing interference constraint, which can considerably limit the secondary network throughput. In this regard, FDRS can offer an adequate solution to boost the secondary throughput while satisfying the imposed interference limit. We first analyze the performance of opportunistic FDRS with residual selfinterference (RSI) by deriving the exact cumulative distribution function of its end-to-end signal-to-interference-plus-noise ratio under Nakagami-m fading. We also evaluate the offered diversity gain of relay selection for different full-duplex cooperation schemes in the presence/absence of a direct source-destination link under Rayleigh fading. When the RSI link gain model is sublinear in the relay power, which agrees with recent research findings, we show that remarkable diversity can be recovered even in the presence of an interfering direct link. Second, we evaluate the end-to-end performance of FDRS with interference constraints due to the presence of a primary receiver. Finally, the presented theoretical findings are verified by numerical simulations.