Trotsky worked under Lenin during and after the Bolshevik Revolution in Russia and Trotskyism is a variant of Marxism-Leninism. It is often contrasted with Stalinism with a major ideological difference being that Stalin officially advocated building Communism in the Soviet Union while Trotsky advocated supporting Communist revolutions in other countries.
Viktor Suvorov and others have argued that Stalin's official position was simply a ruse in order to fool the non-Communist countries (which likely otherwise would have had to classify the Soviet Union as openly supporting violence and revolutions in other countries) and that Stalin covertly also always supported spreading revolutionary Communism elsewhere and even from the start was covertly planning for a future World War of conquest for Communism. The Black Book of Communism, "Part II Word Revolution, Civil War, and Terror", describes extensive covert support by Stalin's Soviet Union for Communist revolutionary activities in other countries, even in the interwar years, contrary to official claims. See also Soviet offensive plans controversy.
Trotskyism was and is one of the most influential forms of Communism outside of the Communist states (where it was banned).
One reasons for this is that Trotskyists often blames all the problems with the Communist states on Stalin, claiming that he corrupted Communism, and ignoring that these problems already existed during the time period when Lenin and Trotsky were in power. See also the articles on Communism and Mass killings under Communist regimes.
Another reason for the popularity may be that Trotsky was Jewish which may have increased interest for Trotskyism among influential Communist Jews. See also Jews and Communism.
Entryism, trying to take over or influence other, often larger, organizations, such as by Trotskyists infiltrating these organizations, is a semi-official part of Trotskyism. There are numerous examples of this by Trotskyists. Trotskyist organizations have also often created front organizations, often officially in order to support a politically correct cause such as "anti-racism" and in coalition with more mainstream organizations, but with the Trotskyists in control or having a large influence.
The more openly pro-revolutionary ideology of Trotskyism may be attractive to the more extreme and violent far leftists such as antifa groups/individuals. They often claim to be Trotskyist or social anarchist.
A characteristic of Trotskyism, possibly related to those attracted to it and the often close associations with anarchist organizations, has been extensive internal strife and sectarianism with there often existing numerous small, mutually hostile, Trotskyist organizations in the same area.