  The problem of quantum equivalence between non-linear sigma models related by
Abelian or non-Abelian T-duality is studied in perturbation theory. Using the
anomalous Ward identity for Weyl symmetry we derive a relation between the Weyl
anomaly coefficients of the original and dual theories. The analysis is not
restricted to conformally invariant backgrounds. The formalism is applied to
the study of two examples. The first is a model based on SU(2) non-Abelian T
duality. The second represents a simple realization of Poisson-Lie T duality
involving the Drinfeld double based on SU(2). In both cases quantum T duality
is established at the 1-loop level.
