In this thesis we investigate free-surface instabilities which occur in various industrial processes involving liquid metal. Specifically, we examine a number of simple problems with a view to developing a plausible theoretical description based on MHD. Of particular interest is the behaviour of the free surface of a pool of liquid metal when it is submitted to an alternating magnetic field. The first problem we examine considers the effect of a low-frequency mag- netic field on a pool of liquid metal. An initially circular pool is deformed into radially oscillating starfish modes at certain critical magnetic field strengths. We study these azimuthal modes theoretically by investigating the behaviour of an isolated mode. We also consider the influence of geometry in our studies of a related problem invloving a rectangular strip of liquid metal. At certain critical magnetic field strengths, an elongated pool is deformed into transverse modes. Aside from complicated mode coupling, the behaviour of these trans- verse modes is similar to the starfish . Both the starfish and transverse modes are caused by a Mathieu-type subharmonic instability. We next consider the effect of a medium-frequency magnetic field on an initially circular pool of liquid metal. Experimentally we study the effect of a vertical alternating medium-frequency magnetic field on an initially circular pool. We observe two types of behaviour: slow radial oscillation of the pool perimeter and rotation of the pool about its centre. The physics of this problem are significantly more complicated than the starfish and strip problems. Accordingly a description in terms of the previous theory is not possible. The final problem we consider is the effect of a high-frequency magnetic field the surface of a conductor. Using a phase-field approach we consider two simple problems: the first is to calculate the rest shape of an infinite strip of liquid resting on a substrate in a vertical gravitational field; the second is to calculate the equilibrium shape of the cross-section of a column of a liquid conductor submitted to a high-frequency magnetic field whose field lines are parallel far from the conductor. Our numeric solutions compare well with previously known analytic solutions.