My research activity is directed toward the mathematical modelling of various fluid mechanics problems. Recently, I have focused on gravity-driven flows of fluid films with a free surface which play an important part in many industrial processes. Such flows are subject to hydrodynamic instability which can result in the appearance of large amplitude wave structures propagating along the surface. In some circumstances the emergence of interfacial instability is an undesirable occurrence. In coating applications, for example, interfacial instability can cause an uneven distribution of material. In other industrial sectors however, film flows exhibiting surface waves can optimize the process. Mass and heat exchangers, for example, operate more efficiently if the surface area of the liquid-gas interface which facilitates the transport is increased. A mathematical model of the flow can be used to investigate the onset of instability in a steady flow and the evolution of the subsequent flow pattern. The nature of the substrate (the solid surface over which the fluid flows) is an important factor affecting the stability of the flow and thus must be included into the model. Furthermore, incorporating into the mathematical formulation such complexities as topography, permeability and applied heating provides a more realistic representation of situations occurring in applications.