Supercomputers notwithstanding, the solution of time-dependent partial differential equations requires the use of adaptive techniques for which the computational mesh is suitably chosen to capture special solution features. Given the wide array of available techniques, it can be a daunting task for users to determine which one might work best for their particular applications.

In this talk, we focus on two basic types of methods of moving the mesh in time and discuss how they encompass many of the approaches which have been used. The history of their development is a long one, with marked practical and theoretical improvements in the methods made by scientists and engineers from a diversity of fields (including many "pure" areas of mathematical analysis). The first approach is based upon minimizing a suitable variational form involving the mesh transformation itself, while the second involves computing mesh velocities directly. Interpreting them both as ways of finding a coordinate transformation from physical to computational coordinates provides insight into why each faces different difficulties for higher dimensional problems. We discuss some recent theoretical developments which help to both explain how these traditional difficulties may be partly overcome. We also relate these adaptive mesh problems to some other general problems in science and engineering. Finally, some numerical examples are given which illustrate the practicability of these new approaches.