  Let $\Omega$ be a compact Hausdorff space, let $E$ be a Banach space, and let
$C(\Omega, E)$ stand for the Banach space of all $E$-valued continuous
functions on $\Omega$ under supnorm. In this paper we study when nuclear
operators on $C(\Omega, E)$ spaces can be completely characterized in terms of
properties of their representing vector measures. We also show that if $F$ is a
Banach space and if $T:\ C(\Omega, E)\rightarrow F$ is a nuclear operator, then
$T$ induces a bounded linear operator $T^\#$ from the space $C(\Omega)$ of
scalar valued continuous functions on $\Omega$ into $\slN(E,F)$ the space of
nuclear operators from $E$ to $F$, in this case we show that $E^*$ has the
Radon-Nikodym property if and only if $T^\#$ is nuclear whenever $T$ is
nuclear.
