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''.

LaTeX2e, 18p. Exposition reworked and slightly compressed; we added a table with a comparison of odd Poisson geometry with Riemannian and even Poisson cases. Latest update: minor editing