  This article is devoted to a review of some recent results on existence,
symmetry and symmetry breaking of optimal functions for
Caffarelli-Kohn-Nirenberg (CKN) and weighted logarithmic Hardy (WLH)
inequalities. These results have been obtained in a series of papers in
collaboration with M. del Pino, S. Filippas, M. Loss, G. Tarantello and A.
Tertikas. Here we put the highlights on a symmetry breaking result: extremals
of some inequalities are not radially symmetric in regions where the symmetric
extremals are linearly stable. Special attention is paid to the study of the
critical cases for (CKN) and (WLH).
