It is well known that poorly shaped elements in a finite element mesh affect both the accuracy of the finite element solution and the stability of the process by which it is computed. Adaptive mesh refinement is one cause for the generation of highly distorted elements. In this talk we will go over some of the mesh improvement techniques such as edge swapping, node insertion, node deletion etc ... We will concentrate on node relocation, also known as mesh smoothing, where each mesh vertex v is recursively moved to a target location that minimizes a given mesh quality measure. The target location vt is expressed as a solution to a local optimization problem that is usually solved using a line search type approach. We will describe an alternative way for solving the local optimization problem by characterizing the set of optimal locations in terms of the level sets associated with the shape quality measure in use. Modulo some assumptions on the shape quality measure, satisfied by most one in use, we completely characterize the set of optimal locations. As a consequence we are able to write explicit formulas for the optimal locations, provided the level sets are simple enough such as circles in 2D and spheres in 3D. Finally we will introduce a new shape measure for tetrahedra.