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Assessing the similarity among 3D shapes is a very complex and challenging research topic. While human perception have been widely studied and produced theories that received a large consensus, the computational aspects of 3D shape retrieval and matching have been only recently addressed. The majority of the methods proposed in the literature mainly focus on the geometry of shapes, in the sense of considering its spatial distribution or extent in the 3D space. From a practical point of view, the main advantage of these methods is that they do not make specific assumption on the topology of the digital models, usually triangle meshes or even triangle soups. Moreover, these methods are also computationally efficient. There is a growing consensus, however, that shapes are recognized and coded mentally in terms of relevant parts and their spatial configuration, or structure. Methods approaching the problem from a geometric point of view do not take into account the structure of the shape and generally the similarity distance between two objects depends on their spatial embedding. The presentation will discuss the definition and use of structural descriptions for assessing shape similarity. The idea is to define a shape description framework based on results of differential topology which deal with the description of shapes by means of the properties of one, or more, real-valued functions defined over the shape. Studying these properties, several topological descriptions of the shape can be defined, which may also encode different geometric and morphological attributes that globally and locally describe the shape. Examples and results will be discussed and ongoing work outlined. This work is partially supported by the EU Newtwork of Excellence AIM{@}SHAPE.