The Critical State model (CSM) is examined for infinite cylindrical samples of a hard type-II superconductor subjected to a transverse oblique magnetic field. Our solution is based on a generalization of a result for the surface current density on a cylindrical surface that produces a uniform interior magnetic field. We give arguments that lead to this generalization. This result has enabled us to get an analytical formulation of the CSM, for cylinders of arbitrary cross section, in the form of an infinite system of first order nonlinear differential equations. An important new outcome of the application of an oblique field is that the current densities +/-Jc within the sample are separated not by a line along the applied field direction as one would naively expect, but along a curve that substantially deviates from it. Following an approximation procedure, we obtain the virgin magnetization curves for a superconducting elliptical cylinder for different orientations of the applied magnetic field. Hysteresis loops for parallel and perpendicular components of magnetization are presented for the applied field orientation of 15°.