The general theory of ionizing shock waves in a magnetic field has been constructed. The theory takes into account precursor ionization of a neutral gas ahead of the shock wave front, caused by photo-ionization, as well as by the impact ionization with electrons accelerated by a transverse electric field induced by the shock front in the incident flow of a neutral gas. The concept of shock wave ionization stability, being basic in the theory of ionizing shock waves in a magnetic field, is introduced. An additional equation for the electric field in the shock wave is obtained. This equation, together with the investigation of the singular point in the downstream flow behind the shock wave front, provides all the information required for solving the problem. For example, this provides two additional boundary conditions for the shock waves of type 2, determining the value and direction of the electric field in the incident flow. One additional boundary condition determines a relation between the value and direction of the electric field for supersonic shock waves of type 3. There are no additional boundary conditions for supersonic shock waves of type 4. The electric field ahead of the shock front has two degrees of freedom. As well as for shocks of other types, its value is less than that of the transverse electric field at which an ionization wave could be emitted by the shock wave front (the ionization stability condition). The additional relationship for supersonic waves of type 4 determines the onset of an isomagnetic (viscous) jump in the structure of the shock wave front. The boundary conditions and ionizing shock wave structures, considered earlier by the authors of the present paper in the ‘limit of electrostatic breakdown’, as well as the structural determination of the electric field, considered earlier by Leonard, are limiting cases in the theory developed here. The ionizing shock wave structures are shown to transform from the GD regime at a low shock velocity to the MHD regime at an enhanced intensity of the shock wave. The abruptness of such a transition (e.g. the transition width on the Mach number scale) is determined by precursor photo-ionization.