  We give a detailed and improved presentation of our recently proposed
formalism for non-linear perturbations in cosmology, based on a covariant and
fully non-perturbative approach. We work, in particular, with a covector
combining the gradients of the energy density and of the local number of
e-folds to obtain a non-linear generalization of the familiar linear uniform
density perturbation. We show that this covector obeys a remarkably simple
conservation equation which is exact, fully non-linear and valid at all scales.
We relate explicitly our approach to the coordinate-based formalisms for linear
perturbations and for second-order perturbations. We also consider other
quantities, which are conserved on sufficiently large scales for adiabatic
perturbations, and discuss the issue of gauge invariance.
