The equations governing the motion of a magnetic uid are presented. These equations are non-linear and give rise to non-Newtonian e ects attributable to the magnetization of the uid. The equations are made dimensionless and the physical parameters of the coupled hydrodynamic -magnetic problem identiΓΏed. The study is ΓΏrst applied to describe the motion of a magnetic droplet freely suspended in a viscous uid undergoing a permanent magnetic ΓΏeld. A ΓΏrst-order theory is developed for the regime of small drop deformation in which viscous forces dominate inertial hydrodynamic force. At this regime, it is shown that the drift velocity of a magnetic drop scales with the square of the applied magnetic ΓΏeld and the deformation of the drop scales linearly with the applied ΓΏeld. Experiments are carried out and the range of validity of the small deformation analysis determined. The pressure-driven ow of a magnetic uid is solved by a regular asymptotic expansion for two cases: a Poiseuille ow of a single magnetic uid and a core pipe ow with a magnetic uid adjacent to the tube wall. The theory is used to predict the volume rate of a viscous magnetic uid separated from a non-magnetic viscous uid by the action of a magnetic ΓΏeld. The apparent viscosity of a magnetic uid as a function of magnetic parameters is also examined from our theory. A possible application of the present theoretical studies is on the remediation technology addressed to oil spills in natural environments.