The bar derived category of a curved dg algebra

Curved A∞-algebras appear in nature as deformations of dg algebras. We develop the basic theory of curved A∞-algebras and, in particular, curved dg algebras. We investigate their link with a suitable class of dg coalgebras via the bar construction and produce Quillen model structures on their module categories. We define the analogue of the relative derived category for a curved dg algebra.