Electromagnetic stirring of liquid metal is widely used during continuous casting in iron and steel industries to improve the quality of the products. In this thesis we aim to develop a theory for traveling linear stirrers in closed channels. We consider a channel filled with an incompressible, electrically conducting viscous fluid. A two dimensional multi Fourier-component magnetic field is moved along the channel. Induced currents in the fluid interact with the field to give a Lorentz force which drives fluid motion. When the magnetic Reynolds number is large the imposed magnetic field is changed significantly. In this study we consider uniformly-moving and accelerating fields. For the initial fluid motion due to a two Fourier-component source moving at a constant velocity, with Rm small, an analytical solution is derived. The LaxWendroff numerical scheme is used to calculate the flow until a steady state is reached. For effective stirring the width of the channel has to be less than the field wavelength. The problem of a solid conductor in a steadily-moving field at finite Rm is investigated next. It is shown that as the magnetic Reynolds number increases the flux inclines in the direction of the motion. In the study of the fluid flow at finite Rm it is shown that the narrower the channel the better and stronger will be the magnetic field's penetration. Most effective stirring occurs when magnetic Reynolds number is around 100. When the magnetic field is given a time-dependent velocity at large Rm we show that more eddies are created immediately after a velocity reversal. In the process, the layers of magnetic fieldlines on the boundaries incline in the opposite direction of the applied field. Simple harmonic motion of the field is also effective, and becomes more so as the frequency increases (at fixed wavelength) but at the expense of overall fluid velocity.