In this talk we report on some mathematical techniques for modelling evolving geometries at low Reynolds numbers. Two problems will be discussed, both involving free capillary surfaces. The first is a study of organisms swimming in Stokes flows in the presence of free surfaces. An idealized mathematical model is presented whereby the swimmer's interaction with a free capillary surface is captured. The second problem is of industrial importance involving the optimal design of thin optic fibres with microstructure. There is much interest in reducing transmission loss in optic fibres by careful design of the microstucture imparted to a fibre during the ``drawing process'' in which molten glass is pulled through a casting die. During this process, geometrical changes in the microstucture take place owing to capillary effects resulting in the need to understand a highly nonlinear inverse problem. New ideas for modelling this process will be described.