Structured mesh quality optimization methods are extended to optimization of unstructured triangular, quadrilateral, and mixed finite element meshes. New interpretations of well-known nodally based objective functions are made possible using matrices and matrix norms. The matrix perspective also suggests several new objective functions. Particularly significant is the interpretation of the Oddy metric and the smoothness objective functions in terms of the condition number of the metric tensor and Jacobian matrix, respectively. Objective functions are grouped according to dimensionality to form weighted combinations. A simple unconstrained local optimum is computed using a modified Newton iteration. The optimization approach was implemented in the CUBIT mesh generation code and tested on several problems. Results were compared against several standard element-based quality measures to demonstrate that good mesh quality can be achieved with nodally based objective functions.