Abstract: In this article, we characterize the solution space of low-degree, implicitly defined, algebraic surfaces which interpolate and/or least-squares approximate a collection of scattered point and curve data in three-dimensional space. The problem of higher-order interpolation and least-squares approximation with algebraic surfaces under a proper normalization reduces to a quadratic minimization problem with elegant and easily expressible solutions. We have implemented our algebraic surface-fitting algorithms, and included them in the distributed and collaborative geometric environment SHASTRA. Several examples are given to illustrate how our algorithms are applied to algebraic surface design.