  This paper deals with symmetry phenomena for solutions of the Dirichlet
problem involving semilinear PDEs on Riemannian domains. We shall present a
rather general framework where the symmetry problem can be formulated and
provide some evidence that this framework is completely natural by pointing out
some results for stable solutions. The case of manifolds with density, and
corresponding weighted Laplacians, is inserted in the picture from the very
beginning.
