Recent work by a number of researchers has highlighted areas in which conservative numerical methods give poor solutions. One such situation is in the modelling of material interfaces. A number of methods for overcoming this shortfall of conservative numerical methods are developed. The flow situations that are considered include multicomponent gases and systems of gases and liquids. It is shown that the errors associated with conservative methods when applied to model gas-liquid interfaces are considerably larger than those for gas-gas interfaces. The first approach used for overcoming the errors in conservative methods is a hybrid primitive-conservative method. This method is used in conjunction with a number of new Riemann solvers for a liquid ambient to provide accurate solutions to a number of challenging one and two dimensional test problems. These test problems include the interaction of a shock wave with a bubble in a gas and an underwater explo.; ion. The application of these hybrid methods to the problem of the interaction of a shock wave with a gas bubble in aa liquid demonstrate that they are unable to provide an accurate solution. Two one dimensional methods are described that are able to provide solutions to such test problems. These methods are the moving grid-Chimera approach and a cut cell approach. The cut cell approach is extended into two dimensions and is shown to be able to provide solutions to the problem of the interaction of a shock wave with a gas bubble in a liquid. This method is also shown to be able to provide more accurate solutions to multicomponent gas problems than those on a standard Cartesian grid.