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

About Existence, Symmetry and Symmetry Breaking for Extremal Functions of Some Interpolation Functional Inequalities