  A Batalin-Vilkovisky formalism is most general framework to construct
consistent quantum field theories. Its mathematical structure is called {\it a
Batalin-Vilkovisky structure}. First we explain rather mathematical setting of
a Batalin-Vilkovisky formalism. Next, we consider deformation theory of a
Batalin-Vilkovisky structure. Especially, we consider deformation of
topological sigma models in any dimension, which is closely related to
deformation theories in mathematics, including deformation from commutative
geometry to noncommutative geometry. We obtain a series of new nontrivial
topological sigma models and we find these models have the Batalin-Vilkovisky
structures based on a series of new algebroids.

Symplectic Geometry;High Energy Physics - Theory