Here is proved uniqueness and existence of the classical solution of a nonlinear Goursat problem with integral boundary conditions. This main result of the paper is obtained on the basis of the contraction mapping principle and the method of successive approximations. A priori estimates for the solution in question and its partial derivatives are found. In addition, the similar Goursat problem is considered for the multi-point boundary conditions instead of the integral ones. References, 6 in number, reflect the specific topic.