  The issue of symmetry and symmetry breaking is fundamental in all areas of
science. Symmetry is often assimilated to order and beauty while symmetry
breaking is the source of many interesting phenomena such as phase transitions,
instabilities, segregation, self-organization, etc. In this contribution we
review a series of sharp results of symmetry of nonnegative solutions of
nonlinear elliptic differential equation associated with minimization problems
on Euclidean spaces or manifolds. Nonnegative solutions of those equations are
unique, a property that can also be interpreted as a rigidity result. The
method relies on linear and nonlinear flows which reveal deep and robust
properties of a large class of variational problems. Local results on linear
instability leading to symmetry breaking and the bifurcation of non-symmetric
branches of solutions are reinterpreted in a larger, global, variational
picture in which our flows characterize directions of descent.
