This paper proposes a rational B-spline hypervolume that represents a volume object which has multiple attributes defined in a multidimensional space. This representation provides a mathematical framework for modeling and visualizing a multidimensional multivariate object as well as analyzing the object interiors to extract its intrinsic features that are directly inaccessible. We discuss the NURBS extension procedure showing that the proposed hypervolume is a generalized volume function not depending on the domain dimensionality and its range dimensionality. Useful expressions arising in connection with a computational treatment are presented for geometric and mathematical analysis of a volume object based on the proposed hypervolume. We also describe the approximation and interpolation algorithms of the proposed hypervolume. Finally, we show various applications such as grid generation, flow visualization, implicit surface modeling, and image morphing. They demonstrate the usefulness and the extensibility of the proposed hypervolume.