For the numerical simulation of many problems of engineering interest, it is desirable to have an automated mesh adaption tool. This is important especially for problems characterized by anisotropic features and require mesh clustering in the direction of high gradients. Another significant issue in meshing emerges in unsteady simulations with moving boundaries, where the boundary motion has to be accommodated by deforming the computational grid. Similarly, there exist problems where current mesh needs to be adapted to get more accurate solutions. To solve these problems, we propose three novel procedures. In the first part of this work, we present an optimization procedure for three-dimensional anisotropic tetrahedral grids based on metric-driven h-adaptation. Through the use of topological and geometrical operators, the mesh is iteratively adapted until the final mesh minimizes a given objective function. We propose an optimization process based on an ad-hoc application of the simulated annealing technique, which improves the likelihood of removing poor elements from the grid. Moreover, a local implementation of the simulated annealing is proposed to reduce the computational cost. Many challenging unsteady multi-physics problems are characterized by moving boundaries and/or interfaces. When the boundary displacements are large, degenerate elements are easily formed in the grid such that frequent remeshing is required. We propose a new r-adaptation technique that is valid for all types of elements (e.g., triangle, tet, quad, hex, hybrid) and deforms grids that undergo large imposed displacements at their boundaries. A grid is deformed using a network of linear springs composed of edge springs and a set of virtual springs. The virtual springs are constructed in such a way as to oppose element collapsing. Both frequent remeshing, and exact-pinpointing of clustering locations are great challenges of numerical simulations, which can be overcome by adaptive meshing algorithms. Therefore, we conclude this work by defining a novel mesh adaptation technique where the entire mesh is adapted upon application of a force field in order to comply with the target mesh or to get more accurate solutions. The method has been tested for two-dimensional problems of a-priori metric definitions as well as for oblique shock clusterings.