We will discuss in which sense bialgebroids and (left) Hopf algebroids can be seen as a notion that unifies various (co)homology theories such as Hochschild, Lie algebroid (in particular Lie algebra and Poisson), or group and etale groupoid (co)homology. We then pass to so-called higher structures on these (co)homoogy groups like cup and cap products, Gerstenhaber algebras or those of Batalin-Vilkovisky modules, i.e., of a noncommutative differential calculus that, among other things, comes along with a contraction, a Lie derivative and a differential. As an illustration, we show how the well-known corresponding operators from differential geometry in the classical Cartan homotopy formula can be obtained from this general approach.