There's a stereotype of the old anarchist terrorist with his bomb. Of course an anarchist fired the shot that directly lead to World War One, and there were anarchist bombings in the US and Europe. How common were anarchist acts of terrorist, how serious a problem was it, and what happened to the anarchists? Were their anarchist movements in non-Western countries? I haven't heard of much about the anarchist movement up until it morphed (or was reborn) as the right-wing US libertarians.
Would it be fair to say that early 20th century fascism was in part a response to anarchist terrorism? Or was 19th/20th century anarchist terrorism a response to authoritarian governments?
(I hope that I won't violate the 20 year rule by inviting comparisons to modern Islamic terrorism.)
American anarchism is more of my area of expertise, so I'll mostly answer you from that point of view.
To begin with, how are you defining anarchism? Those that fired the shots that led to World War I were Serbian Nationalists, not anarchists. Not all acts of terrorism in the 19th and 20th century were by anarchists (most, in fact, were not)
Anarchists acts of violence were largely overblown by the media of the day. In America, at any rate Margaret S. Marsh asserts that there were only three cases of propaganda by the deed (terrorism) in the United States. First came Alexander Berkman’s attempt to assassinate Henry Clay Frick, an attempt planned with Emma Goldman, which ultimately failed in its desired outcome. Second was the assassination of President William McKinley by Leon Czolgosz, whose ties to the anarchists were tenuous at best, and third, the Lexington Avenue bomb explosion of 1914, which killed one woman and three male anarchists. Accordingly, “In all, counting the seven policemen killed at Haymarket, anarchist terrorism in the United States was responsible for the deaths of twelve people, three of them anarchists.” (Margaret S. Marsh, Anarchist Women, 8.). In America, at least, Anarchism was not a serious problem, though they were convenient scapegoats.
As for what happened to anarchists in America: some were executed (Haymarket, Sacco and Vanzetti), some were deported (Goldman and Berkman), some were harassed (Lucy Parsons), some dropped out of the movement, some became Communists. The movement was ever changing. I could go into more detail if you'd like, but for now I'll move on.
There were indeed anarchist movements in non-Western countries. There was Shūsui Kōtoku and the High Treason Incident of Japan. China had anarchists, one of whom was Ba Jin, a famous author. There were anarchist groups in S. America, notably Mexico.
As for anarchist morphing or being reborn, that is patently untrue. There is still anarchism with anarchists, not Libertarians. This is, however, a very complicated answer, since I'm sure, like other movements, members broke of and did their own thing. I would not attribute the rise of right-wing libertarianism to the Anarchists- if anything they were much more likely to become Communists than move to a right wing group. Anarchism is very much still alive, though it seems, struggling to find its political footing.
I do not think it would be fair to say that fascism was a partial response to anarchist terrorism simply because that is giving the anarchist movements and terroristic actions more power than they actually possessed. In America anarchist terrorism was in large part a response to anti-union activities on the part of robber barons and the police. Simple answer, but when people feel that governments do not allow them legitimate ways to deal with their grievances, they will turn to other methods. However, if you read Goldman's autobiography or work by Parsons, De Cleyre, Tucker, et al. the anarchist movement was in large part move theoretically oriented than action oriented.