This project is about using moving meshes - r-adaptivity - to improve the predictive power of atmospheric flow simulations, which are used in the fields of numerical weather prediction and climate modelling. When the atmosphere is simulated on a computer, this is done by dividing the sphere into cells which are arranged in a mesh. There is a conflict between the need for accuracy, which requires smaller (and hence more) cells, and computational efficiency, which increases with the number of cells. A reasonable question to ask is: for a given amount of accuracy, what size of cells do I need? The answer can be provided mathematically, but it depends on what is actually happening in the atmosphere simulation. Much smaller cells are required in the regions of smaller scale features such as atmospheric fronts, cyclones, jets, convective cells etc. It then seems like a waste to choose the same cell size all over the globe even in regions where these features are absent. An attractive idea is to try to stretch, deform and move the mesh around so that smaller cells are used in the regions of small scale features, and larger cells are used elsewhere. This would mean that a better compromise can be made between accuracy and computational efficiency, thus improving predictive power for the same resource. This idea has been used successfully in many engineering applications, and the goal of this project is to transmit this technology to atmosphere simulation, where it can be used by meteorologists and climate scientists to take their science forward. There are, however, a number of challenging aspects. Efficient mesh movement algorithms have not previously been developed for the sphere geometry which is needed for global atmosphere simulations. There is the question of how to detect where the mesh should be moved to. It is also the case that it is very challenging to design stable and accurate numerical algorithms for simulating the atmosphere, and these must be adapted to remain stable and accurate under mesh movement. All of these questions and issues will be addressed in this project.