  We consider polynomial maps $f:\C\to\C$ of degree $d\ge 2$, or more generally
polynomial maps from a finite union of copies of $\C$ to itself which have
degree two or more on each copy. In any space $\p^{S}$ of suitably normalized
maps of this type, the post-critically bounded maps form a compact subset
$\cl^{S}$ called the connectedness locus, and the hyperbolic maps in $\cl^{S}$
form an open set $\hl^{S}$ called the hyperbolic connectedness locus. The
various connected components $H_\alpha\subset \hl^{S}$ are called hyperbolic
components. It is shown that each hyperbolic component is a topological cell,
containing a unique post-critically finite map which is called its center
point. These hyperbolic components can be separated into finitely many distinct
``types'', each of which is characterized by a suitable reduced mapping schema
$\bar S(f)$. This is a rather crude invariant, which depends only on the
topology of $f$ restricted to the complement of the Julia set. Any two
components with the same reduced mapping schema are canonically biholomorphic
to each other. There are similar statements for real polynomial maps, or for
maps with marked critical points.
