  For associative algebras in many different categories, it is possible to
develop the machinery of Gr\"obner bases. A Gr\"obner basis of defining
relations for an algebra of such a category provides a "monomial replacement"
of this algebra. The main goal of this article is to demonstrate how this
machinery can be used for the purposes of homological algebra. More precisely,
our approach goes in three steps. First, we define a combinatorial resolution
for the monomial replacement of an object. Second, we extract from those
resolutions explicit representatives for homological classes. Finally, we
explain how to "deform" the differential to handle the general case. For
associative algebras, we recover a well known construction due to Anick. The
other case we discuss in detail is that of operads, where we discover
resolutions that haven't been known previously. We present various
applications, including a proofs of Hoffbeck's PBW criterion, a proof of
Koszulness for a class of operads coming from commutative algebras, and a
homology computation for the operads of Batalin--Vilkovisky algebras and of
Rota--Baxter algebras.
