Many important problems in computational science and engineering involve partial differential equations posed on moving domains. This talk will present numerical methods for the solution of such problems, as well as theoretical tools for analyzing their accuracy. I will first introduce a family of high-order finite element methods for moving-boundary problems that can handle large domain deformations with ease while representing the geometry of the moving domain exactly. At the core of our approach is the use of a universal mesh: a background mesh that contains the moving domain and conforms to its geometry at all times by perturbing a small number of nodes in a neighborhood of the moving boundary. I will then introduce a unified analytical framework for establishing the convergence properties of a wide class of numerical methods for moving-boundary problems. This class includes, as special cases, the technique described above as well as conventional deforming-mesh methods (commonly known as arbitrary Lagrangian-Eulerian, or ALE, schemes).