  Gian-Carlo Rota suggested in one of his last articles the problem of
developing a theory around the notion of integration algebras, complementary to
the already existing theory of differential algebras. This idea was mainly
motivated by Rota's deep appreciation for Kuo-Tsai Chen's seminal work on
iterated integrals. As a starting point for such a theory of integration
algebras Rota proposed to consider a particular operator identity first
introduced by the mathematician Glen Baxter. Later it was coined Rota-Baxter
identity. In this article we briefly recall basic properties of Rota--Baxter
algebras, and present a concise review of recent work with a particular
emphasis of noncommutative aspects.
