I will discuss the algebraic structure of homology theories defined by a left Hopf algebroid U 
over a possibly noncommutative base algebra A, such as for example Hochschild, Lie algebroid 
(in particular Lie algebra and Poisson), or group and étale groupoid (co)homology: 
Ext_U(A,A) is a Gerstenhaber algebra that acts via a generalised Lie derivative and cap product 
on Tor^U(M,A), and there is a generalised Cartan homotopy formula that relates the Lie derivative 
to cyclic homology. As an application, Ginzburg's result that the cohomology ring of a Calabi–Yau 
algebra is a Batalin–Vilkovisky algebra is extended to twisted Calabi–Yau algebras. 

(joint work with Niels Kowalzig)