Abstract: The complexity of 3D shapes that are represented in digital form and processed in CAD/CAM/CAE, entertainment, biomedical, and other applications has increased considerably. Much research was focused on coping with or on reducing shape complexity. However, what exactly is shape complexity? We discuss several complexity measures and the corresponding complexity reduction techniques. Algebraic complexity measures the degree of polynomials needed to represent the shape exactly in its implicit or parametric form. Topological complexity measures the number of handles and components or the existence of non-manifold singularities, non-regularized components, holes or self-intersections. Morphological complexity measures smoothness and feature size. Combinatorial complexity measures the vertex count in polygonal meshes. Representational complexity measures the footprint and ease-of-use of a data structure, or the storage size of a compressed model. The latter three vary as a function of accuracy.