In this lecture I report on essentially two results for overdetermined boundary value problems and the p -Laplace operator. The first one is joint work with H. Shahgholian on Bernoulli type free boundary problems that model for instance galvanization processes. For this family of problems the limits p ® ¥ and p ® 1 lead to interesting analytical and surprising geometric questions. In particular for the case p ® 1 I add more recent results, that are not contained in [12]. The second one is joint work with F. Gazzola and I. Fragala [6]. It provides an alternative and more geometric proof of Serrin's seminal symmetry result for positive solutions to overdetermined boundary value problems. As a byproduct I give an analytical proof for the geometric statement that a closed plane curve of curvature not exceeding K must enclose a disk of radius 1/ K .