In this survey we consider the classical overdetermined problem which was
studied by Serrin in 1971. The original proof relies on Alexandrov's moving
plane method, maximum principles, and a refinement of Hopf's boundary point
Lemma. Since then other approaches to the same problem have been devised. Among
them we consider the one due to Weinberger which strikes for the elementary
arguments used and became very popular. Then we discuss also a duality approach
involving harmonic functions, a shape derivative approach and a purely integral
approach, all of them not relying on maximum principle. For each one we
consider pros and cons as well as some generalizations
