We analyze the joint efforts of the geometry processing and numerical analysis communities in the last decades to define and measure the concept of “mesh quality”. Researchers have been striving to determine how, and how much, the accuracy of a numerical simulation or a scientific computation (e.g., rendering, printing, modeling) depends on the mesh adopted to model the problem, and which geometrical features most influence the result. We overview the most common quality indicators currently used to evaluate the goodness of a discretization and drive mesh generation, coarsening, and refinement processes. We analyze indicators defined over two- and three-dimensional meshes with any type of element (triangular/tetrahedral, quadrangular/hexahedral, and generic polygonal/polyhedral). We also present an algorithm for optimizing polygonal and polyhedral meshes using these quality indicators, which analyzes local element quality and agglomerates elements to optimize global mesh quality, with a parameter controlling the percentage of elements to be removed. This significantly reduces the number of degrees of freedom associated with a discrete problem defined over the mesh, particularly for high-order formulations, and regularizes low-quality meshes by removing pathological elements. In practical applications to constrained domains and time-dependent problems, we show how the convergence rate is preserved in the optimized meshes, and the approximation errors are comparable with those obtained with the original ones, despite the drastic reduction of degrees of freedom. We showcase the functionalities of the open-source tool PEMesh2, which allows to compute and compare mesh quality using different indicators, optimize meshes, and solve a numerical problem, comparing optimizations driven by different quality indicators. The notion of mesh quality and the optimization algorithm are discussed in the context of evolving geometries arising in moving mesh schemes, where preserving mesh quality under deformation is essential for accuracy, stability, and computational efficiency, and preliminary results are presented.