  Curved algebras are algebras endowed with a predifferential, which is an
endomorphism of degree -1 whose square is not necessarily 0. This makes the
usual definition of quasi-isomorphism meaningless and therefore the homotopical
study of curved algebras cannot follow the same path as differential graded
algebras.
  In this article, we propose to study curved algebras by means of curved
operads. We develop the theory of bar and cobar constructions adapted to this
new notion as well as Koszul duality theory. To be able to provide meaningful
definitions, we work in the context of objects which are filtered and complete
and become differential graded after applying the associated graded functor.
  This setting brings its own difficulties but it nevertheless permits us to
define a combinatorial model category structure that we can transfer to the
category of curved operads and to the category of algebras over a curved operad
using free-forgetful adjunctions.
  We address the case of curved associative algebras. We recover the notion of
curved Aoo-algebras, and we show that the homotopy categories of curved
associative algebras and of curved Aoo-algebras are Quillen equivalent.
