In the first part of this talk, I discuss the generation of meshes adapted to a prescribed scalar 'monitor' function. This is done through equidistribution, so that the volume of a cell is inversely proportional to the monitor function. We supplement this with an optimal transport condition, which aids with mesh regularity, and guarantees existence and uniqueness of such a mesh. The resulting mesh can be obtained by solving a Monge-Ampère equation, a scalar nonlinear elliptic PDE. This optimal transport also approach generalizes naturally from Euclidean space to manifolds such as the sphere. In the second part of this talk, I discuss the integration of moving mesh adaptivity into a finite element shallow water model, in the wider context of the need for global numerical weather prediction models that can resolve small-scale dynamic features. We do this by modifying the governing fluid equations so they are solved in a frame relative to the moving mesh. The finite element discretization is based on a 'compatible', or 'mimetic', approach, in which the finite element spaces are linked by differential operators. The degrees of freedom correspond not just to point values, but also to fluxes and densities, which complicates the modifications that are required.