This paper concerns the practical applications and modelling of Stefan problems with a flowing thin liquid layer. The modelling will be discussed in the context of two practically important scenarios, Leidenfrost (when a liquid droplet floats above a hot surface) and contact melting. The governing equations will be derived and then reduced to a more tractable system. Along the way we will introduce an accurate variant of the Heat Balance Integral Method, which allows us to approximate the solution to the Stefan problem on a finite domain. In both cases excellent agreement between the model results and experimental data will be demonstrated.