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In cognitive radio (CR) systems, reliable spectrum sensing techniques are required in order to avoid interference to the primary users of the spectrum. Whereas most of the existing literature on spectrum sensing considers impairment by additive white Gaussian noise (AWGN) only, in practice, CRs also have to cope with various types of non-Gaussian noise such as man-made impulsive noise, co-channel interference, and ultra-wideband interference. In this paper, we propose an adaptive Lp-norm detector which does not require any a priori knowledge about the primary user signal and performs well for a wide range of circularly symmetric non-Gaussian noises with finite moments. We analyze the probabilities of false alarm and missed detection of the proposed detector in Rayleigh fading in the low signal-to-noise ratio regime and investigate its asymptotic performance if the number of samples available for spectrum sensing is large. Furthermore, we consider the deflection coefficient for optimization of the Lp-norm parameters and discuss its connection to the probabilities of false alarm and missed detection. Based on the deflection coefficient an adaptive algorithm for online optimization of the Lp-norm parameters is developed. Analytical and simulation results show that the proposed Lp-norm detector yields significant performance gains compared to conventional energy detection in non-Gaussian noise and approaches the performance of the locally optimal detector which requires knowledge of the noise distribution.