  We investigate the dynamics of semigroups generated by a family of polynomial
maps on the Riemann sphere such that the postcritical set in the complex plane
is bounded. The Julia set of such a semigroup may not be connected in general.
We show that for such a polynomial semigroup, if $A$ and $B$ are two connected
components of the Julia set, then one of $A$ and $B$ surrounds the other. From
this, it is shown that each connected component of the Fatou set is either
simply or doubly connected. Moreover, we show that the Julia set of such a
semigroup is uniformly perfect. An upper estimate of the cardinality of the set
of all connected components of the Julia set of such a semigroup is given. By
using this, we give a criterion for the Julia set to be connected. Moreover, we
show that for any $n\in \Bbb{N} \cup \{\aleph_{0}\} ,$ there exists a finitely
generated polynomial semigroup with bounded planar postcritical set such that
the cardinality of the set of all connected components of the Julia set is
equal to $n.$ Many new phenomena of polynomial semigroups that do not occur in
the usual dynamics of polynomials are found and systematically investigated.
