We present numerical simulations that include 1-D Eulerian multi-group radiationhydrodynamics, 1-D non-Local-Thermodynamic-Equilibrium (non-LTE) radiative transfer, and 2-D polarised radiative transfer for super-luminous interacting supernovae (SNe). Our reference model is a ∼10 M ⊙ inner shell with 10 51 erg ramming into a ∼3 M ⊙ cold outer shell (the circumstellar-medium, or CSM) that extends from 10 15 to 2×10 16 cm and moves at 100 km s −1 . We discuss the light curve evolution, which cannot be captured adequately with a grey approach. In these interactions, the shock-crossing time through the optically-thick CSM is much longer than the photon diffusion time. Radiation is thus continuously leaking from the shock through the CSM, in disagreement with the shell-shocked model that is often invoked. Our spectra redden with time, with a peak distribution in the near-UV during the first month gradually shifting to the optical range over the following year. Initially Balmer lines exhibit a narrow line core and the broad line wings that are characteristic of electron scattering in the SNe IIn atmospheres (CSM). At later times they also exhibit a broad blue shifted component which arises from the cold dense shell. Our model results are broadly consistent with the bolometric light curve and spectral evolution observed for SN 2010jl. Invoking a prolate pole-to-equator density ratio in the CSM, we can also reproduce the ∼2% continuum polarisation, and line depolarisation, observed in SN 2010jl. By varying the inner shell kinetic energy and the mass and extent of the outer shell, a large range of peak luminosities and durations, broadly compatible with super-luminous SNe IIn like 2010jl or 2006gy, can be produced.