Notch signalling is widely used throughout development in the determination of cell fates and maintenance of progenitors in many developmental systems. In this thesis, nonlinear ordinary differential equation models and discrete delay differential equation models of the Notch signalling pathway are investigated mathematically to understand the dynamics of cell-fate determination. We use linear stability analysis to find conditions for when oscillatory dynamics can be observed in bistable systems. We compare how this can be achieved for each type of model, and demonstrate how this affects the temporal dynamics of the decision-making process. The models are then extrapolated to a larger population scale to understand how the size and geometry of the population can affect the rate at which cells can determine their fate, and the ratio of alternate cell types. We also show conditions for when stable global oscillations can exist without bistability. Finally, we use vertex-based modelling to introduce Notch signalling into a proliferating population of cells, to demonstrate how the timescales of proliferation and cell-fate determination interact. Specifically, we show that both the rate of proliferation and the presence of oscillatory dynamics can affect the rate of differentiation, supporting results in current literature.