  We compute the fully non-linear Cosmic Microwave Background (CMB)
anisotropies on scales larger than the horizon at last-scattering in terms of
only the curvature perturbation, providing a generalization of the linear
Sachs-Wolfe effect at any order in perturbation theory. We show how to compute
the n-point connected correlation functions of the large-scale CMB anisotropies
for generic primordial seeds provided by standard slow-roll inflation as well
as the curvaton and other scenarios for the generation of cosmological
perturbations. As an application of our formalism, we compute the three- and
four-point connected correlation functions whose detection in future CMB
experiments might be used to assess the level of primordial non-Gaussianity,
giving the theoretical predictions for the parameters of quadratic and cubic
non-linearities f_NL and g_NL.
