The direct and inverse scattering problems related to objects having inhomogeneous impedance boundaries are addressed by considering cylindrical bodies. In the solution of the direct scattering problem, the scattered field is first expressed in terms of a combined single- and double-layer potential through Green's formula and the boundary condition. By using the jump relations on the boundary of the object, the scattering problem is reduced to a boundary integral equation that can be solved via a Nystrom method. The aim of the inverse impedance problem is to reconstruct the inhomogeneous surface impedance of the body from the measured far field data. Here representing the scattered field as a single- layer potential leads to an ill-posed integral equation of the first kind for the density that requires stabilization for its numerical solution; for example, by Tikhonov regularization. With the aid of the jump relations the single- layer potential enables the evaluation of the total field and its derivative on the boundary of the scatterer. Consequently, from the boundary condition the surface impedance can be reconstructed either by direct evaluation or by a minimum norm solution in the least squares sense. The numerical results show that our methods yields good resolution both for the direct and the inverse problem.