  A commuting $n$-tuple $(T_1, \ldots, T_n)$ of bounded linear operators on a
Hilbert space $\clh$ associate a Hilbert module $\mathcal{H}$ over
$\mathbb{C}[z_1, \ldots, z_n]$ in the following sense: \[\mathbb{C}[z_1,
\ldots, z_n] \times \mathcal{H} \rightarrow \mathcal{H}, \quad \quad (p, h)
\mapsto p(T_1, \ldots, T_n)h,\]where $p \in \mathbb{C}[z_1, \ldots, z_n]$ and
$h \in \mathcal{H}$. A companion survey provides an introduction to the theory
of Hilbert modules and some (Hilbert) module point of view to multivariable
operator theory. The purpose of this survey is to emphasize algebraic and
geometric aspects of Hilbert module approach to operator theory and to survey
several applications of the theory of Hilbert modules in multivariable operator
theory. The topics which are studied include: generalized canonical models and
Cowen-Douglas class, dilations and factorization of reproducing kernel Hilbert
spaces, a class of simple submodules and quotient modules of the Hardy modules
over polydisc, commutant lifting theorem, similarity and free Hilbert modules,
left invertible multipliers, inner resolutions, essentially normal Hilbert
modules, localizations of free resolutions and rigidity phenomenon.
  This article is a companion paper to "An Introduction to Hilbert Module
Approach to Multivariable Operator Theory".
