Modeling free-form curves and surfaces is one of the fundamental problems in computer aided geometric design. To solve this problem, several modeling techniques have been proposed. Three of these techniques, are investigated. The unifying theme of these three techniques is the use and the control of geometric continuity. The first technique deals with constructing parametric spline curves with controlled continuity between the spline segments at the knots. An axiomatic approach to geometric continuity for parametric representations is proposed. Based on this totally algebraic approach, many new flexible notions of continuity are developed. Corresponding to these notions, new spline curves are constructed in a way that gives the designer more control over the curve shape. Many examples are given. When derivative information is available Hermite interpolation can be used to build high continuity surfaces. A dynamic programming algorithm that solves the problem of interpolating bivariate Hermite data where the interpolation positions are aligned on a triangular grid is developed and analyzed. The third geometric continuity problem arises when modeling with subdivision surfaces, in reducing the continuity of these surfaces to allow for the insertion of sharp edges/vertices on these surfaces. A new approach to solving this problem is introduced and analyzed with illustrative examples.