The formulation of a local cell quality metric [Branets, L., and Carey, G. F., 2003, Proceedings of the 12th International Meshing Roundtable, Santa Fe, NM, pp. 371–378;Engineering with Computers (in press)] for standard elements defined by affine maps is extended here to the case of elements with quadratically curved boundaries. We show for two-dimensional and three-dimensional simplex elements with quadratically curved boundaries that all cases of map degeneracy can be identified by the metric. Moreover, we establish a “maximum principle” which allows estimating the bounds on the quality metric. The nondegeneracy conditions for biquadratic quadrilaterals with one curved edge are also determined. The metric is implemented in an untangling/smoothing algorithm for improving unstructured meshes including simplex elements that have curved boundary segments. The behavior and efficiency of this algorithm is illustrated for numerical test problems in two and three dimensions.