The implicit complex model allows for the representation of heterogeneous objects as multidimensional point sets with multiple attributes, where elements of different dimensions and with attributes of different nature are combined together in a topological structure. In this chapter, we present in detail the underlying function-based model for a certain class of cells to be used among several others in the implicit complex model. Each cell is dimensionally homogeneous and defined by a real vector-function. We provide a brief survey of different modelling techniques related to point sets with attributes. It spans such different areas as solid modelling, scalar fields, volume models, and material modelling. Then, on the basis of this survey we formulate requirements to a more general model. In the presented generalizing constructive hypervolume model, point set geometry and attributes are represented independently using real-valued scalar functions with their underlying tree data structures. While 3D and higher dimensional entities have been widely studied, we present function-based definitions of lower dimensional entities, such as surface patches and curve segments, with a corresponding trimming technique. A high level language supporting modelling function-based cells and attributes is described and illustrative examples are provided.