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This paper studies averaging algebras, say, associative algebras endowed withaveraging operators. We develop a cohomology theory for averaging algebras andjustify it by interpreting lower degree cohomology groups as formaldeformations and abelian extensions of averaging algebras. We make explicit the$L_\infty$-algebra structure over the cochain complex defining cohomologygroups and introduce the notion of homotopy averaging algebras as Maurer-Cartanelements of this $L_\infty$-algebra.

Cohomology theory of averaging algebras, $L_\infty$-structures and homotopy averaging algebras