Numerical methods for partial differential equations based on adaptive meshes have been proposed since the eighties, first by covering several theoretical settings (from linear to nonlinear problems, from steady to unsteady settings), and successively becoming instrumental to a huge variety of applicative fields. In particular, numerical discretizations based on adapted computational grids allowed to attain results which were out of range with traditional computational tools. Different toolboxes may be required to generate an adaptive discretization of the computational domain: • Mesh modification mechanisms, possibly driven by an error assessment; • A posteriori metric-based mesh generation driven by robust error estimates; • Strategic implementation confirming an evident reduction of the global computational effort; • Goal-oriented adaptivity and/or optimal mesh generation; • Conservative interpolation from mesh to mesh; • Parallel computing with remeshing and repartitioning of the whole computational effort. The MS aims at gathering researchers working in the field of adaptive grids, to offer a rich panorama on the state of the art and, at the same time, significative instances of application. The topics of interest for the MS range from computable error estimators and academic benchmarks to industrial applications, such as multiphase and multiphysics problems, turbulence modeling, topology optimization, image analysis, predictive medicine, design of materials, aerospace and automotive prototyping, uncertainty quantification.