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The problem of twelve spheres is to understand, as a function of $r \in(0,r_{max}(12)]$, the configuration space of $12$ non-overlapping equal spheresof radius $r$ touching a central unit sphere. It considers to what extent, andin what fashion, touching spheres can be varied, subject to the constraint ofalways touching the central sphere. Such constrained motion problems are ofinterest in physics and materials science, and the problem involves topologyand geometry. This paper reviews the history of work on this problem, presentssome new results, and formulates some conjectures. It also presents generalresults on configuration spaces of $N$ spheres of radius $r$ touching a centralunit sphere, with emphasis on $3 \le N \le 14$. The problem of determining themaximal radius $r_{max}(N)$ is a version of the Tammes problem, to whichL\'aszl\'o Fejes T\'oth made significant contributions.

Configuration Spaces of Equal Spheres Touching a Given Sphere: The Twelve Spheres Problem