  A method is developed to construct irreducible representations(irreps) of the
quantum supergroup $U_q(C(n+1))$ in a systematic fashion. It is shown that
every finite dimensional irrep of this quantum supergroup at generic $q$ is a
deformation of a finite dimensional irrep of its underlying Lie superalgebra
$C(n+1)$, and is essentially uniquely characterized by a highest weight. The
character of the irrep is given. When $q$ is a root of unity, all irreps of
$U_q(C(n+1))$ are finite dimensional; multiply atypical highest weight irreps
and (semi)cyclic irreps also exist. As examples, all the highest weight and
(semi)cyclic irreps of $U_q(C(2))$ are thoroughly studied.
