We consider the following overdetermined problem Δu = f(u),u > 0 in Ω u = 0 on (e)Ω (e)vu = C on (e)Ω Serrin proved(1971)that if the domain is bounded then the domain must be a ball and that u must be radially symmetric.This talk is concerned with the case when Ω is unbounded.In 1997,Berestycki,Caffarelli and Nirenberg conjectured that if Ω is connected and Serrins overdetermined problem admits a solution then Ω is either a half space,or a cylinder,or the complement of a ball.In this talk we present results of BCN conjecture in both positive and negative answers.We first show that if the domain Ω is an epigraph then Ω must be an half space,provided that one of the following assumptions are satisfied: either n = 2,or the graph is globally Lipschitz; or n ≤ 8 and (e)u/(e)vn>0 in Ω.Then we show that this result is optimal by constructing an epigraph in dimensions n ≥9,which is a perturbation of Bombieri-De Giorgi-Giusti minimal graph,such that Serrins overdetermined problem admits a solution.Furthermore we establish an intricate relation between Serrins overdetermined problem and CMC surfaces.We prove that for any nondegenerate CMC surfaces or minimal surfaces Serrins overdetermined problem is solavble.