Studying extreme weather events, such as heatwaves, is a fascinating growing field and poses several methodological and computational challenges. Understanding the physical mechanisms which drive heatwaves and being able to forecast them is of pivotal importance in the context of mitigation and adaptation to the current climate crisis. A new theoretical framework which addresses both tasks is developed in this thesis and applied to mid-latitudes heatwaves. This framework, called the Gaussian Approximation, relies on simple but meaningful assumptions on the statistics of weather fields relevant for heatwaves. It proves to capture well the salient features of atmospheric conditions predominant during those events and to have competitive or even better predictive skills than other more complex weather prediction models. Another major challenge when dealing with extreme heatwaves, and extreme events in general, is that they suffer from an intrinsic sampling problem, due to their rareness. Historical records are too short and direct simulations are simply too expensive for having a good estimation of these events. Sampling algorithms, such as the rare events algorithms, tackle this task resulting in both improving the quality and the efficiency of extreme events simulation. In this thesis, this class of algorithm has been applied for the study of double jet stream state, an interesting rare phenomenon which is linked to heatwave occurrence at mid-latitudes. The study consists in coupling a climate model (CESM) with a properly chosen score function, which describes the salient features of the phenomenon, in order to sample extreme and unprecedented events.