Summary: In this article, we discuss the discontinuous oblique derivative boundary value problem for nonlinear uniformly elliptic systems of second order equations in multiply connected domains. We first propose the discontinuous oblique derivative problem and its new modified well-posedness. Next we give a priori estimates of solutions of the modified discontinuous boundary value problem for the corresponding elliptic system of first order complex equations and verify its solvability by the above estimates of solutions and the Leray-Schauder theorem. Finally the solvability results of the original discontinuous oblique derivative problem can be derived. Here, we mention that the discontinuous boundary value problems possess many applications in mechanics and physics etc.