Hochschild chains of an associative algebra are a noncommutative generalization of differential forms on a manifold. As such, they might be expected to possess an algebraic structure similar to one that exists on forms. They do indeed carry some of it, notably an analog of the De Rham differential, and fail to carry other structure, such as the cup product. We will discuss the (hidden) traces of the product and of a BV structure that they do have. In particular we will define the notion of a Hamiltonian action of a Hopf algebra. The exposition will be based in part on the Ginzburg-Schedler approach to Hochschild and cyclic homology.