The past few decades witnessed a flurry of research activities in the area of Computer-Aided Geometric Design (CAGD), especially with relation to subdivision surfaces, where particular attention is given to the interpolation of curves or network of curves by such surfaces. This paper traces the major research landmarks pursued in this domain to achieve interpolation of points and curves by such surfaces, specifically by means of polygonal complexes. Along the way, the paper touches upon related research directions, especially the additional interpolation of normal vectors and curvatures following the same approach and under the same settings.