Hopf algebras are often regarded as a very technical area of algebra, and non-trivial examples beyond the Drinfeld-Jimbo quantum groups are rarely looked at. (Can you name any mathematical software package which can do Hopf algebra computations?) In representation theory, a Hopf algebra is an object whose representations can be tensor-multiplied. Hence Hopf algebras can act on noncommutative rings, generalising group and Lie algebra actions on various commutative and super-commutative structures. But because the definition of a Hopf algebra is self-dual, one can consider coactions as well as actions --- these have no obvious analogue for groups. The purpose of my talk will be to give an introduction to Hopf algebra (co)actions and then to describe a hitherto obscure construction, known to specialists as
Takeuchi product or a Doi-Koppinen module, which can create a new associative algebra out of two algebras with a (co)action of a Hopf algebra. A result due to A. Berenstein and myself shows that any algebra which factorises into two subalgebras is in fact a Takeuchi product. For example, new Hopf algebras arise from the factorisation in the classical PBW theorem - some of these were discovered by Majid in his thesis; and Lusztig's graded affine Hecke algebra leads to Nichols algebras and answers a question of Etingof. If time permits, I will mention why, in light of the above, the recent work on coideal subalgebras (Heckenberger et al) could be of interest to "mainstream" representation theorists.