The subject of this talk will be associative algebras with inner products, their homotopy versions, and their deformation theory. The homotopy versions of associative algebras with inner products (A_\infty algebras with \infty-inner products) make an appearence in theoretical physics and topology. The deformation theory of A_\infty algebras with \infty-inner products, which is related to cyclic cohomology, and has some familiar features, as such things go, but also has some unique features.