In this chapter we are concerned with mesh generation methods and mesh adaptivity issues. Nowadays, many techniques are available to complete meshes of arbitrary domains for computational purposes. Planar, surface and volume meshing have been automated to a large extent. Over the last few years, meshing activities have focused on adaptive schemes where the features of a solution field must be accurately captured. To this end, meshing techniques must be revisited in order to be capable of completing high quality meshes conforming to these features. Error estimates are therefore used to analyze the solution field at a given stage and, based on the results and the information they yield, adapted meshes are created before computing the next stage of the solution field. A number of novel meshing issues must be addressed including how to construct a mesh adapted to what the error estimate prescribes, how to validate and construct high-order meshes, how to handle large size meshes, how to consider moving boundary problems, etc.