  We build resolutions for general twisted tensor products of algebras. These
bimodule and module resolutions unify many constructions in the literature and
are suitable for computing Hochschild (co)homology and more generally Ext and
Tor for (bi)modules. We analyze in detail the case of Ore extensions,
consequently obtaining Chevalley-Eilenberg resolutions for universal enveloping
algebras of Lie algebras (defining the cohomology of Lie groups and Lie
algebras). Other examples include semidirect products, crossed products, Weyl
algebras, Sridharan enveloping algebras, and Koszul pairs.
