  Recently N. Berkovits, motivated by the supertwistor description of ${\cal
N}=4 D=4$ super Yang-Mills, considered the generalization of the ${\cal N}=1
D=4$ $\theta$-twistor construction to D=10 and applied it for a compact
covariant description of ${\cal N}=1 D=10$ super Yang-Mills. This supports the
relevance of the $\theta$-twistor as a supersymmetric twistor alternative to
the well-known supertwistor. The minimal breaking of superconformal symmetry is
an inherent property of the $\theta$-twistor received from its fermionic
components, described by a Grassmannian vector instead of a Grassmannian scalar
in the supertwistor. The $\theta$-twistor description of the ${\cal N}=1 D=4$
massless chiral supermultiplets $(S, S + 1/2)$ with spins $S=0,1/2,1,3/2,2,...$
is considered here. The description permits to restore the auxiliary $F$ fields
of the chiral supermultiplets absent in the supertwistor approach. The proposed
formalism is naturally generalized to ${\cal N}=4 D=4 $ and can be used for an
off-shell description of the corresponding super Yang-Mills theory.
