  We propose two constructions extending the Chern-Moser normal form to
non-integrable Levi-nondegenerate (hypersurface type) almost CR structures. One
of them translates the Chern-Moser normalization into pure intrinsic setting,
whereas the other directly extends the (extrinsic) Chern-Moser normal form by
allowing non-CR embeddings that are in some sense "maximally CR". One of the
main differences with the classical integrable case is the presence of the
non-integrability tensor at the same order as the Levi form, making impossible
a good quadric approximation - a key tool in the Chern-Moser theory. Partial
normal forms are obtained for general almost CR structures of any CR
codimension, in particular, for almost-complex structures. Applications are
given to the equivalence problem and the Lie group structure of the group of
all CR-diffeomorphisms.
