  In this paper, we study the representation theory of Hopf-Ore extensions of
group algebras and pointed Hopf algebras of rank one over an arbitrary field
$k$. Let $H=kG(\chi, a,\d)$ be a Hopf-Ore extension of $kG$ and $H'$ a rank one
quotient Hopf algebra of $H$, where $k$ is a field, $G$ is a group, $a$ is a
central element of $G$ and $\chi$ is a $k$-valued character for $G$ with
$\chi(a)\neq 1$. We first show that the simple weight modules over $H$ and $H'$
are finite dimensional. Then we describe the structures of all simple weight
modules over $H$ and $H'$, and classify them. We also consider the
decomposition of the tensor product of two simple weight modules over $H'$ into
the direct sum of indecomposable modules. Furthermore, we describe the
structures of finite dimensional indecomposable weight modules over $H$ and
$H'$, and classify them. Finally, when $\chi(a)$ is a primitive $n$-th root of
unity for some $n>2$, we determine all finite dimensional indecomposable
projective objects in the category of weight modules over $H'$.
