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This article examines the quality assessment of planar quadrilateral mesh elements in a comprehensive way. First, an analytic characterization of quadrangular shape is provided, and existing concepts of stretching and skewness, earlier proposed for specific geometries, are generalized. Then, two triangle quality measures are extended to quadrilaterals and their respective extremal and asymptotic behaviors examined, showing in particular that even if needed they cannot detect the triangular degeneracy of a quadrilateral. An existing quality measure is then discussed, which is able to handle this case. In particular, an unbalanced asymptotic behavior is demonstrated, justifying the need for a new approach. Toward this goal, the triangle quality measure based on Frobenius norm is modified in order to replace equilateral reference element by right isosceles triangles, with control on the specific right angle. Finally, two new quadrilateral quality measures are designed and examined using these results. Numerical results illustrate the matter.