Abstract: For any given sequence of points, the expression of cubic Catmull-Rom spline basis function is used to construct a curve through those points, and the sufficient and necessary conditions of cubic Catmull-Rom spline preserving convex interpolation are derived. Furthermore, by using the concept of generalized convexity, the sufficient and necessary conditions of the cubic parameter Catmull-Rom spline preserving generalized convex interpolation are derived. When the given sequence of points satisfies the sufficient and necessary conditions for preserving generalized convex interpolation obtained, the cubic parameter Catmull-Rom spline curve of the interpolating points is an automatically preserving generalized convex and G1 continuous curve. The validity of the method and the correctness of the theory are proved by some constructed examples.