  In discrete geometry, the contact number of a given finite number of
non-overlapping spheres was introduced as a generalization of Newton's kissing
number. This notion has not only led to interesting mathematics, but has also
found applications in the science of self-assembling materials, such as
colloidal matter. With geometers, chemists, physicists and materials scientists
researching the topic, there is a need to inform on the state of the art of the
contact number problem. In this paper, we investigate the problem in general
and emphasize important special cases including contact numbers of minimally
rigid and totally separable sphere packings. We also discuss the complexity of
recognizing contact graphs in a fixed dimension. Moreover, we list some
conjectures and open problems.
