The self-similar solution that characterizes the water impact, with a constant vertical velocity, of a wedge entering the free surface with an arbitrary orientation is derived analytically. The study is carried out by assuming the fluid to be ideal, weightless and with negligible surface tension effects. The solution is based on the complex analysis of nonlinear two-dimensional problems of unsteady free boundary flows and is written in terms of two governing functions, which are the complex velocity and the derivative of the complex potential defined in a parameter domain. The boundary value problem is reduced to the system of an integral and an integro-differential equation in terms of the velocity modulus and of the velocity angle to the free surface, both written as functions of a parameter variable. The system of equations is solved through a numerical procedure which is validated in the case of symmetric wedges. Comparisons with data available in literature are established for this purpose. Results are presented in terms of free surface shape, contact angles at the intersection with the wedge boundary, pressure distribution, force and moment coefficients. For a given wedge angle, the changes induced by the heel angle on the above quantities are discussed. A criterion is proposed to determine the limit conditions beyond which flow separation from the wedge apex occurs. Comparisons with experimental results available in literature are presented.