  Abelian T-duality in Gauged Linear Sigma Models (GLSM) forms the basis of the
physical understanding of Mirror Symmetry as presented by Hori and Vafa. We
consider an alternative formulation of Abelian T-duality on GLSM's as a gauging
of a global $U(1)$ symmetry with the addition of appropriate Lagrange
multipliers. For GLSMs with Abelian gauge groups and without superpotential we
reproduce the dual models introduced by Hori and Vafa. We extend the
construction to formulate non-Abelian T-duality on GLSMs with global
non-Abelian symmetries. The equations of motion that lead to the dual model are
obtained for a general group, they depend in general on semi-chiral
superfields; for cases such as $SU(2)$ they depend on twisted chiral
superfields. We solve the equations of motion for an $SU(2)$ gauged group with
a choice of a particular Lie algebra direction of the vector superfield. This
direction covers a non-Abelian sector that can be described by a family of
Abelian dualities. The dual model Lagrangian depends on twisted chiral
superfields and a twisted superpotential is generated. We explore some
non-perturbative aspects by making an Ansatz for the instanton corrections in
the dual theories. We verify that the effective potential for the $U(1)$ field
strength in a fixed configuration on the original theory matches the one of the
dual theory. Imposing restrictions on the vector superfield, more general
non-Abelian dual models are obtained. We analyze the dual models via the
geometry of their susy vacua.
