  We provide a detailed treatment of the Camassa--Holm (CH) hierarchy with
special emphasis on its algebro-geometric solutions. In analogy to other
completely integrable hierarchies of soliton equations such as the KdV or AKNS
hierarchies, the CH hierarchy is recursively constructed by means of basic
polynomial formalism invoking a spectral parameter. Moreover, we study
Dubrovin-type equations for auxiliary divisors and associated trace formulas,
consider the corresponding algebro-geometric initial value problem, and derive
the theta function representations of algebro-geometric solutions of the CH
hierarchy.
