Freeze-drying is a process extensively used in the pharmaceutical industry as a solution on how to reduce the water content of temperature-sensitive materials and increase their stability and shelf life. However, at the moment, freeze-drying remains the most expensive stage of pharmaceutical manufacturing, and hence further modelling is needed. To model the process, Stefan problems are considered. A numerical method based on the level set approach and compact finite differencing is developed and adapted for solidification scenarios. Various one and two-dimensional solidification problems are considered. These include solidification in a rectangle with Dirichlet and convective flux boundary conditions. To further investigate the behaviour of the model analytically, small time asymptotic solutions have been developed and used to start the numerical computation. The model is later extended to simulate the freezing process on multiple three-dimensional vials with simplified cuboid geometry. The extended model is used to investigate the 'edge vial' effect caused by non-symmetrical heat transfer inside the freeze-drying chamber. The results are presented that show that under certain conditions, the 'edge vial' effect can cause non-uniformity in freezing rates of the edge and corner vials when compared to the centre vials. Lastly, a novel heuristic model of freezing is developed based on dynamics of chemical reactions. The model is investigated analytically and asymptotic solutions are presented in different time scales and compared to full numerical simulations. The results show a good agreement between the asymptotic and numerical solutions.