Alien abduction was the fear of being taken by aliens from one's home planet.
In 2151, Subcommander T'Pol stated alien abduction conspiracies had been present on Earth "for centuries". She suggested that taking an Akaali up to Enterprise NX-01 might spawn a similar meme on the Akaali homeworld. (ENT: "Civilization")
In 2153, Enterprise discovered a group of Humans whose ancestors had been taken from Earth, by the Skagarans, sometime during the 19th century. (ENT: "North Star")
In 2371, the USS Voyager discovered a Human presence on a planet in the Delta Quadrant. It was discovered that the Briori had taken the Humans from Earth in 1937 and utilized them as slave labor. The slaves later revolted and overthrew the Briori, establishing their own community on the planet. (VOY: "The 37's")