We construct dual Lagrangians for GH models in two space-time dimensions for arbitrary Lie Groups G and H ⊂ G. Our approach does not require choosing coordinates on GH, and allows for a natural generalization to Poisson-Lie T-duality. For the case where the target metric on GH is induced from the invariant group metric on G, the dual system is a gauged Higgs model, with a non-constant metric and a coupling to an antisymmetric tensor. The dynamics for the gauge connection is governed by a BF-term. Poisson-Lie T-duality is relevant once we allow for a more general class of target metrics, as well as for couplings to an antisymmetric tensor, in the primary theory. Then the dual theory is written on a group G dual to G, and the gauge group H (which, in general, is not a subgroup of G acts non-linearly on G. The dual system therefore gives a non-linear realization of a gauge theory. All dual descriptions are shown to be canonically equivalent to the corresponding primary descriptions, at least at the level of the current algebra.