  We generalize the BV formalism for the physical theories on supermanifolds
with graded symmetry algebras realized off-shell and on-shell. An application
of such generalization to supersymmetric theories allows us to formulate the
new classification which refines the usual off-shell/on-shell classification.
Our new classification is based on the type of higher order antifield terms in
the corresponding BV action. We provide explicit examples for each class of the
refined classification. We show that the supersymmetric localization for the
off-shell theories is a BV integral over particular Lagrangian submanifold. We
generalize the supersymmetric localization to the on-shell supersymmetric
theories of quadratic type in our classification. The partition function for
such theories becomes Gaussian integral for a particular choice of Lagrangian
submanifold.
