  The purpose of this article is to present a pedagogical review of T-dualityin
in string theory. The evolution of the closed string is envisaged on the
worldsheet in the presence of its massless excitations. The duality symmetry is
studied when some of the spacial coordinates are compactified on d-dimensional
torus, $T^d$. The known results are reviewed to elucidate that equations of
motion for the compact coordinates are $O(d,d)$ covariant, $d$ being the number
of compact directions. Next, the vertex operators of excited massive levels are
considered in a simple compactification scheme. It is shown that the vertex
operators for each massive level can be cast in a T-duality invariant form in
such a case. Subsequently, the duality properties of superstring is
investigated in the NSR formulation for the massless backgrounds such as
graviton and antisymmetric tensor. The worldsheet superfield formulation is
found to be very suitable for our purpose. The Hassan-Sen compactification is
adopted and it is shown that the worldsheet equations of motion for compact
superfields to be very suitable for our purpose. The Hassan-Sen
compactification is adopted and it is shown that the worldsheet equations of
motion for compact superfields are $O(d,d)$ covariant when the backgrounds are
independent of superfields along compact directions. The vertex operators for
excited levels are presented in the NS-NS sector and it is shown that they can
be cast in T-duality invariant form for the case of Hassan-Sen compactification
scheme. An illustrative example is presented to realize our proposal.
