Abstract: This dissertation addresses the problems of building interpolating or approximating implicit surfaces from a heterogeneous collection of geometric primitives like points, polygons, and spline/subdivision surface patches. The user can choose to generate a surface that exactly interpolates the geometric elements, or a surface that approximates the input by smoothing away features smaller than some user-specified size. The implicit functions are represented using a scattered data interpolation formulation known as moving least-squares with constraints at input points or integrated over the parametric domain of each polygon or surface patch. This dissertation also proposes an improved technique for enforcing normal constraints that overcomes undesirable oscillatory behavior produced by previous methods. Multiple points, polygons and surface patches can be blended together by a single implicit function whose isosurface is a manifold envelope that either interpolates or approximates the original input, even when self-intersections, holes, or other defects are present. With an iterative procedure for ensuring that the implicit surface tightly encloses the input elements, the resulting clean, manifold surface can then be used for generating volume meshes for finite elements, manufacturing rapid prototyping models, and other applications that require manifold surfaces.