  We consider odd Laplace operators acting on densities of various weight on an
odd Poisson (= Schouten) manifold $M$. We prove that the case of densities of
weight 1/2 (half-densities) is distinguished by the existence of a unique odd
Laplace operator depending only on a point of an ``orbit space'' of volume
forms. This includes earlier results for odd symplectic case, where there is a
canonical odd Laplacian on half-densities. The space of volume forms on $M$ is
partitioned into orbits by a natural groupoid whose arrows correspond to the
solutions of the quantum Batalin--Vilkovisky equations. We give a comparison
with the situation for Riemannian and even Poisson manifolds. In particular,
the square of odd Laplace operator happens to be a Poisson vector field
defining an analog of Weinstein's ``modular class''.
