This paper is concerned with the problem of reconstructing the surfaces of three-dimensional objects, given a collection of planar contours representing cross-sections through the objects. This problem has important aplications in biomedical research and instruction, solid modeling, and industrial inspection. The method we describe produces a triangulated mesh from the data points of the contours which is then used in conjunction with a piecewise parametric surface-fitting algorithm to produce a reconstructed surface. The problem can be broken into four subproblems: the correspondence problem (which contours should be connected by the surface?), the tiling problem (how should the contours be connected?), the branching problem (what do we do when there are branches in the surface?), and the surface-fitting problem (what is the precise geometry of the reconstructed surface?) We describe our system for surface reconstruction from sets of contours with respect to each of these subproblems. Special attention is given to the correspondence and branching problems. We present a method that can handle sets of contours in which adjacent contours share a very contorted boundary, and we describe a new approach to solving the correspondence problem using a Minimum Spanning Tree generated from the contours.