In this paper we study certain curvature properties of Kenmotsu manifolds with respect to the quarter-symmetric metric connection. First we investigate Weyl projective symmetric Kenmotsu manifolds with respect to the quarter-symmetric metric connection. Next, we study Kenmotsu manifolds satisfying the curvature condition P·S = 0, where P and S are the projective curvature tensor and Ricci tensor respectively with respect to the quartersymmetric metric connection. Further, we discuss about pseudoprojectively flat and φ-projectively semisymmetric Kenmotsu manifolds with respect to the quarter-symmetric metric connection. Finally, we give an example of a 5-dimensional Kenmotsu manifold admitting a quarter-symmetric metric connection for illustration. © 2016, Transilvania University of Brasov 1. All rights reserved.