To enhance the national security, there is a growing need for authorized parties to legitimately monitor suspicious communication links for preventing intended crimes and terror attacks. In this paper, we propose a new wireless information surveillance paradigm by investigating a scenario, where a legitimate monitor aims to intercept a suspicious wireless link over fading channels. The legitimate monitor can successfully eavesdrop (decode) the information of the suspicious link at each fading state only when its achievable data rate is no smaller than that at the suspicious receiver. We propose a new approach, namely, proactive eavesdropping via cognitive jamming, in which the legitimate monitor purposely jams the receiver in a full-duplex mode so as to change the suspicious communication (e.g., to a smaller data rate) for overhearing more efficiently. By assuming perfect self-interference cancelation (SIC) and global channel state information (CSI) at the legitimate monitor, we characterize the fundamental information-theoretic limits of proactive eavesdropping. We consider both delay-sensitive and delay-tolerant applications for the suspicious communication, under which the legitimate monitor maximizes the eavesdropping non-outage probability (for event-based monitoring) and the relative eavesdropping rate (for content analysis), respectively, by optimizing the jamming power allocation over different fading states subject to an average power constraint. Numerical results show that the proposed proactive eavesdropping via cognitive jamming approach greatly outperforms other benchmark schemes. Furthermore, by extending to a more practical scenario with residual SI and local CSI, we design an efficient online cognitive jamming scheme inspired by the optimal cognitive jamming with perfect SIC and global CSI.