  We obtain a complete group classification of the Lie point symmetries of
nonlinear Poisson equations on generic (pseudo) Riemannian manifolds M. Using
this result we study their Noether symmetries and establish the respective
conservation laws. It is shown that the projection of the Lie point symmetries
on $M$ are special subgroups of the conformal group of M. In particular, if the
scalar curvature of M vanishes, the projection on M of the Lie point symmetry
group of the Poisson equation with critical nonlinearity is the conformal group
of the manifold. We illustrate our results by applying them to the Thurston
geometries.
