The first part of this paper deals with the following general question: Given a second order elliptic nonlinear partial differential equation with a certain symmetry (e.g., spherical symmetry), do positive solutions exhibit the same symmetry? This is related to the phenomenon of “symmetry breaking” in physics. For a large class of nonlinearities, we show that positive solutions respect the symmetry of the equation. That positivity is crucial for such a result may be seen from the linear eigenvalue problems, where the lowest eigenvalue has an eigenfunction (the “ground state,” in physical terms) which is strictly positive (or negative) and has the maximum symmetry. Higher eigenfunctions change sign and are not spherically symmetric. Some of the equations we treat are classical approximations to quantum or statistical mechanics systems. A symmetry may not be broken at the classical level, but broken only after quantum or thermodynamic effects are taken into account (here we shall not touch upon quantum or statistical problems). The results we describe here (and in several extensions) were proven in [1]. The proofs are based on certain forms of the maximum principle, and a device of A. D. Alexandroff. These techniques were employed before by Serrin [2] who treated solutions of elliptic equations with over-determined boundary conditions. The results of [1] are 256useful in proving uniqueness theorems, as well as in deriving a priori bounds [3] for positive solutions.