In this paper, we study the existence and nonexistence of nontrivial solutions to the following critical biharmonic problem with the Steklov boundary conditions Δ²=+Δ+||^2**-2 in , =Δ+=0 on , where ,, ∈ , ⊂ ^N( ≥ 5) is a unit ball, 2^** = 2/N-4 denotes the critical Sobolev exponent for the embedding ²() →^2** () and is the outer normal derivative of on . Under some assumptions on , and , we prove the existence of nontrivial solutions to the above biharmonic problem by the Mountain pass theorem and show the nonexistence of nontrivial solutions to it by the Pohozaev identity.