The separating flow of an inviscid fluid is not only a limit solution of the steady separating, laminar fluid flow at high values of the Reynolds number but it is also part of its structure. This work aims at reexamining the separating flow of inviscid fluid past a bluff body which is fixed in an otherwise uniform stream of fluid. For the purpose of this paper we will assume that the bluff body is a circular cylinder but the theory is applicable to bodies of any shape. It is further assumed that the fluid is in steady two-dimensional motion and is inviscid and of constant density. The flow structure is assumed to consist of a separated flow region, cavitating flows in which there exists a free surface on which the pressure is constant, and a wake. A twin spiral vortex model is used in order to determine the shape of the free streamline. Based on the free streamline theory the problem reduces to solving a mixed boundary value problem and a Hilbert solution for the inverse problem in the auxiliary plane is obtained. When we consider the flow in the physical plane the problem is transformed into a direct problem in which the geometry of the solid body is given in advance. We assume that the separation is smooth and thus the curvature of the free streamline at the point of free detachment be equal to that of the body surface. A numerical method for solving the two-dimensional potential flows past arbitrarily shaped curved bluff bodies is developed.