The purpose of this thesis is to present an overview of principal items in the recent research on the geometric portent of the Laplace-Beltrami operator which is
canonically attached to a Riemannian manifold. Frankly an exploded view of an expository paper by R. Brooks supported by closely related articles, this work starts
with an elementary treatment of the isospectrality of flat tori to go an exposition of Sunada's seminal adaptation of a number theoretical idea and ends with a
description of the barest essentials of isospectral deformations on Riemannian manifolds.

Keywords : Laplace-Beltrami operator, spectrum, isospectrality, isospectral deformation, nilmanifold