(also nonabelian homological algebra)
A curved dg-algebra is like a dg-algebra, but instead of the differential squaring to 0, it squares to the graded commutator with a fixed element of the algebra: its “curvature”.
This is like the covariant derivative on the sections of a vector bundle with connection satisfying $\nabla \circ \nabla = F_\nabla$, where $F_\nabla$ is the curvature 2-form of the connection (valued, here, in fiber endomorphism)s.
Curved dg-algebras appear in the description of various TQFTs.
(…)
A basic exposition of the definition is in
For applications in derived categories of D-branes in Landau-Ginzburg models see
Dmitri Orlov, Derived Categories of Singularities and D-branes in Landau-Ginzburg models , (arXiv:math.ag/0503632)
Anatoly Preygel, Thom-Sebastiani and duality for matrix factorizations arXiv:1101.5834
An natural construction of curved dg-algebras as de Rham / Dolbeault complexes on a circle 2-bundle with connection is in
and with more details in section 2 of