We consider slow steady flows of a conducting fluid at large values of the Hartmann number and small values of the magnetic Reynolds number in an inhomogeneous magnetic field. The general solution is obtained in explicit form for the basic portion (core) of the flow, where the inertia and viscous forces may be neglected. The boundary conditions which this solution must satisfy at the outer edges of the boundary layers which develop at the walls are considered. Possible types of discontinuity surfaces and other singularities in the flow core are examined. An exact solution is obtained for the problem of conducting fluid flow in a tube of arbitrary section in an inhomogeneous magnetic field. The content of this paper is a generalization of some results on flows in a homogeneous magnetic field, obtained in [1–8], to the case of arbitrary flows in an inhomogeneous magnetic field. The author's interest in the problems considered in this study was attracted by a report presented by Professor Shercliff at the Institute of Mechanics, Moscow State University, in May 1967, on the studies of English scientists on conducting fluid flows in a strong uniform magnetic field.