  Let $M$ be a complex manifold which admits an exhaustion by open subsets
$M_j$ each of which is biholomorphic to a fixed domain $\Omega \subset \mathbb
C^n$. The main question addressed here is to describe $M$ in terms of $\Omega$.
Building on work of Fornaess--Sibony, we study two cases namely, $M$ is
Kobayashi hyperbolic and the other being the corank one case in which the
Kobayashi metric degenerates along one direction. When $M$ is Kobayashi
hyperbolic, its complete description is obtained when $\Omega$ is one of the
following domains -- (i) a smoothly bounded Levi corank one domain, (ii) a
smoothly bounded convex domain, (iii) a strongly pseudoconvex polyhedral domain
in $\mathbb C^2$, or (iv) a simply connected domain in $\mathbb C^2$ with
generic piecewise smooth Levi-flat boundary. With additional hypotheses, the
case when $\Omega$ is the minimal ball or the symmetrized polydisc in $\mathbb
C^n$ can also be handled. When the Kobayashi metric on $M$ has corank one and
$\Omega$ is either of (i), (ii) or (iii) listed above, it is shown that $M$ is
biholomorphic to a locally trivial fibre bundle with fibre $\mathbb C$ over a
holomorphic retract of $\Omega$ or that of a limiting domain associated with
it. Finally, when $\Omega = \Delta \times \mathbb B^{n-1}$, the product of the
unit disc $\Delta \subset \mathbb C$ and the unit ball $\mathbb B^{n-1} \subset
\mathbb C^{n-1}$, a complete description of holomorphic retracts is obtained.
As a consequence, if $M$ is Kobayashi hyperbolic and $\Omega = \Delta \times
\mathbb B^{n-1}$, it is shown that $M$ is biholomorphic to $\Omega$. Further,
if the Kobayashi metric on $M$ has corank one, then $M$ is globally a product;
in fact, it is biholomorphic to $Z \times \mathbb C$, where $Z \subset \Omega =
\Delta \times \mathbb B^{n-1}$ is a holomorphic retract.
