The paper deals with the notions «information extremism», «information terrorism», their classification is provided, the forms of manifestation are proposed. The purpose is to find out the specifics of these notions. The study is urgent because subjects of information terrorism and extremism having various mass media at their disposal can impose their attitudes and ideas on the masses in different countries of the world. Information extremism is often considered together with PR-technologies, propaganda and other kinds of linguistic manipulation of the audience. As for information extremism, there is only one general aim, i. e. obtaining power over people's minds, recruiting new members to the extremist organization or sending the masses into a panic. It is seen as a preparation stage for other types of extremism, providing favorable conditions for distribution and acceptance of extremist ideas and ideals. Information terrorism is implemented in the field including different attitudes and ideas: political, philosophical, legal, esthetic, religious etc., i. e. in the area of the battle of ideas. Dozens of thousands of today's websites and web-portals are actively used by extremists and terrorists. Having accounts in social networks they can recruit new followers and conduct propaganda of radicalism more easily. Information extremism and terrorism in all their manifestations should be classified as enormously destructive crimes with significant influence on human minds and society. The efforts made by governments are not always efficient to monitor and neutralize websites used by extremists and terrorists. The struggle against information extremism and terrorism is a priority of the world leaders. To overcome this evil international cooperation is required.