  We formulate $\lambda$-deformed $\sigma$-models as QFTs in the upper-half
plane. For different boundary conditions we compute correlation functions of
currents and primary operators, exactly in the deformation parameter $\lambda$
and for large values of the level $k$ of the underlying WZW model. To perform
our computations we use either conformal perturbation theory in association
with Cardy's doubling trick, as well as meromorphicity arguments and a
non-perturbative symmetry in the parameter space $(\lambda,k)$, or standard QFT
techniques based on the free field expansion of the $\sigma$-model action, with
the free fields obeying appropriate boundary conditions. Both methods have
their own advantages yielding consistent and rich, compared to those in the
absence of a boundary, complementary results. We pay particular attention,
albeit not exclusively, to integrability preserving boundary conditions.
