Atmospheric in-flight ice accretion has been a significant operational hazard in aviation for decades. Super-cooled water droplets impinge on exposed surfaces such as wings and rotor blades. These droplets may freeze on the surface thereby changing lift characteristics and disturb weight and aerodynamic balances. The multiple length scales involved prevent designing dynamically similar flows making traditional aeronautical engineering tools such as wind tunnel experiments not suitable. Therefore, computational fluid dynamics (CFD) methods have proved an attractive alternative to study atmospheric icing effects. However, most approaches are based on simple incompressible models and are only suited for small ice heights due to the difficulty of dynamically tracking the ice accretion. This thesis aims to develop novel mathematical models to capture more relevant phenomena and to improve the numerical methods to allow dynamic tracking of the air-ice interface. The initial chapter presents an augmented air and droplet model which tracks droplet temperatures thereby producing more accurate heat fluxes for the phase transition calculation. Firstly, we validate our novel model for common ice accretion test cases and find excellent agreement with literature. The advantage of the augmented system is demonstrated by applying it to an experimental setup that studies the heat exchange between water droplets and air for various flow conditions. We find excellent agreement between our model and the experiment for all presented cases whereas constant-temperature approaches match only for short interaction times. Finally, we apply the new system to study the droplet temperatures around various aerofoil and find significant temperature differences compared with conventional models. The following chapter studies the freezing process on the wing geometry. Presently, the most advanced model is based on lubrication theory, however, linear terms are truncated. We extend the series expansion to include first order terms and demonstrate that the additional order is necessary to accurately capture the thin film flow on a cylinder. Furthermore, we extend the lubrication-theory- based approach which was limited to simple geometries. The extended model is valid on arbitrary wing shapes making it more relevant for engineers studying real-world problems. The penultimate chapter combines the previous two to give a simulation of the full icing process. We integrate it with a Cartesian cut-cell method which can cope dynamically with moving interfaces. The robustness and performance of the cut-cell techniques allow us to simulate ice growth on real-world geometries. We demonstrate this capability by presenting results of the dynamic ice growth on a NACA 0012 aerofoil - making this the first such numerical experiment.