The article analyses the reasons for the trends of extremism and terrorism committed via the Internet. The research is made of the forms of the Internet use for terrorist and extremist purposes, in particular, conspiratorial communication between members of these communities, the recruitment of new members, remote control of such criminal communities and collection of funds for criminal purposes. Analysis is given of the problem of the Internet use by criminals to study the public opinion and the psychology of people with social intolerance. The description is given to the trends of criminal activity under study; analysis is made of improving the methods of its implementation. The author comes up with the judgment about the need for a system of forensic measures to prevent such crimes, based on the legal, organizational, technical and methodological measures. A specialized program of crime prevention is offered. The article describes the main activities to be included in it, in particular, identification, fixing and blocking materials of extremist nature. To clarify the mechanisms of such offenses, their trace picture, the identity of these criminals, and other forensically relevant questions, the offer is made to develop and implement the Honeypot research system.