Permafrost is a component of Earth's cryosphere which is of importance for ecosystems and infrastructure in the Arctic, and plays a key role in the global carbon cycle. Global climate warming which is particularly pronounced in polar regions constitutes a major disturbance to permafrost environments which rely on a vulnerable thermal equilibrium between the atmosphere and the land surface. Large-scale climate models reveal high uncertainties in projections of how much permafrost would thaw in response to climate warming scenarios, since they do not represent key complexities of permafrost environments such as small-scale landscape heterogeneities and feedbacks through lateral transport processes. In particular, large-scale models do not take into account thaw processes in ice-rich permafrost which cause widespread landscape change referred to as thermokarst. For this thesis, I have developed a numerical model to investigate thaw processes in ice-rich permafrost landscapes, and I have used it to obtain improved projections of how much permafrost would thaw in response to climate warming. The focus of my research was on cold, ice- and carbon-rich permafrost deposits in the northeast Siberian Arctic, and on landscapes characterized by ice-wedge polygons. In three closely interrelated research articles I have demonstrated that the novel modelling approach of laterally coupled ``tiles'' can be used to realistically simulate the evolution of ice-rich permafrost landscapes. The numerical simulations have revealed that small-scale lateral transport of heat, water, snow, and sediment crucially affect the dynamics of permafrost landscapes and how much permafrost would thaw under climate warming scenarios. My research revealed that substantially more permafrost carbon is affected by thaw in numerical simulations which take into account thermokarst processes, than in simulations which lack a representation of excess ice. These results suggest that conventional large-scale models used for future climate projections might considerably underestimate permafrost thaw and associated carbon-cycle feedbacks. Overall, the research presented in this thesis constitutes a major progress towards the realistic assessment of ice-rich permafrost landscape dynamics using numerical models, and demonstrates the high potential of tile-based modelling paradigms for the computationally efficient representation of subgrid-scale heterogeneity and lateral processes in large-scale climate models.