This article surveys the field of Artificial Intelligence for theories of abstraction. We identify several unifying themes and argue that, even though these theories make different representational choices, they are all trying to capture the same intuitive ideas. Progress has also been made in understanding the reduction in search brought by abstraction. In this paper, we demonstrate how these theories of abstraction have helped comprehend the advantages abstraction provides. Finally, several different systems with abstraction have been implemented, tested and used. We focus on how the theories of abstraction have been influenced by and have themselves influenced the development of such systems.