The problem of reconstructing smooth surfaces from discrete data, such as a triangular mesh, is an important practical problem in geometrical design, medical imaging, computer graphics and computer vision. Most approaches attempt to stitch together small polynomial surface patches along their boundaries. We will describe an alternative approach inspired by the overlapping of coordinate charts on a surface. A major advantage of this "manifold-based" approach is that it splits the problem of enforcing smoothness of overlapping charts (using the gluing data, reflecting the topology of the mesh) and the problem of finding parametrizations (using the geometry of the mesh). We will indicate how to make this method practical, and explain its advantages.