This paper contains a theoretical treatment of the axisymmetric stagnation point problem in magnetohydrodynamics. Two distinct arrangements of the magnetic field are considered, with the lines of force lying in the plane of, and perpendicular to the streamlines, respectively. The boundary conditions are examined carefully and the solutions are completed by joining them to meaningful distributions of magnetic field in the solid region. Exact solutions can be obtained by numerical methods, but it is more illuminating to consider approximations valid for very large and very small diffusion numbers (ratio of kinematic viscosity to magnetic diffusivity). Distinct boundary layers of current and vorticity occur in these limiting cases, which can be treated separately. For flows with the first arrangement of the magnetic field, and in which the field does not vanish at infinity, it is deduced that the solution refers to a forward stagnation point when the Alfven number is less than unity, and to a rear stagnation point when the Alfven number exceeds unity. An interesting choking effect is found in flows with the second arrangement of the magnetic field when the diffusion number is small, and the Alfven number exceeds unity; it is believed that this effect could be demonstrated in the laboratory.