  We define the concept of higher order differential operators on a general
noncommutative, nonassociative superalgebra A, and show that a vertex operator
superalgebra has plenty of them, namely modes of vertex operators. A linear
operator \Delta is a differential operator of order at most r if an inductively
defined (r+1)-form \Phi_{\Delta}^{r+1} is identically zero. When A is
supercommutative and associative and \Delta is an odd, square zero, second
order differential operator on A, \Phi_{\Delta}^2 defines a "Batalin-Vilkovisky
algebra" structure on A. We generalize this notion to any superalgebra with an
odd, square zero, second order differential operator, and show that all
properties of the classical BV bracket but one continue to hold "on the nose".
We also point out connections to Leibniz algebras and the noncommutative
homology theory of Loday. Taking the generalization process one step further,
we remove all conditions on \Delta to examine the changes in the basic
properties of the bracket. We see that a topological chiral algebra with a mild
restriction yields a classical BV algebra in the cohomology. Finally, we
investigate the quantum BV master equation and relate it to deformations of
differential graded algebras.
