A differential graded (dg) manifold (M,Q) is a sheaf R of Z-graded commutative algebras together with a derivation Q of degree +1 whose square is equal to zero. The sheaf R is thought of as the sheaf of smooth functions on the (virtual) "graded space" M and the derivation Q as a "homological" vector field on M. Many geometric and algebraic structures can be described as dg manifolds, for instance: Lie algebras, complex manifolds, regular foliations, and L_oo algebras.

Using Kontsevich's famous formality theorem, Liao, Xu and myself established a formality theorem for smooth dg manifolds: given any finite-dimensional dg manifold (M,Q), there exists an L_oo quasi-isomorphism of differential graded Lie algebras from an appropriate space of polyvector fields endowed with the Schouten bracket [-,-] and the differential [Q,-] to an appropriate space of polydifferential operators endowed with the Gerstenhaber bracket [[-,-]] and the differential [[m+Q,-]], whose first Taylor coefficient (1) is equal to the composition of the action of the square root of the Todd class of the dg manifold (M,Q) on the space of polyvector fields with the Hochschild--Kostant--Rosenberg map and (2) preserves the associative algebra structures on the level of cohomology. The Todd class of dg manifolds extends both the classical Todd class of complex manifolds and the Duflo element of Lie theory.