Abstract. The general theory of self-similar magnetohydrodynamic (MHD) expansion waves is presented. Building on the familiar hydrodynamic results, a complete range of possible field–flow and wave-mode orientations are explored. When the magnetic field and flow are parallel, only the fast-mode wave can undergo an expansion to vacuum conditions: the self-similar slow-mode wave has a density that increases monotonically. For fast-mode waves with the field at an arbitrary angle with respect to the flow, the MHD equations have a critical point. There is a unique solution that passes through the critical point that has ½γβ = 1 and Br = 0 there, where γ is the polytropic index, β the local plasma beta and Br the radial component of the magnetic field. The critical point is an umbilical point, where sound and Alfvén speeds are equal, and the transcritical solution undergoes a change from a fast-mode to a slow-mode expansion at the critical point. Slow-mode expansions exist for field-flow orientations where the angle between field and flow lies either between 90° and 180° or between 270° and 360°. There is also an umbilic point in these solutions when the initial plasma beta β0 exceeds a critical value βcrit. When β0 [ges ] βcrit, the solutions require a transition through a critical point. When β0 < βcrit, there is a smooth solution involving an inflection in the density and angular velocity. For other angles between field and flow, all the slow-mode waves are compressive. An analytic solution for the case of a magnetic field everywhere perpendicular to the flow with γ = 2 is presented.