  In this manuscript we consider the transformations of the oscillators of the
bosonic fields of the superstring in terms of the fermions oscillators and vice
versa. We demand the exchange of the commutation and anti-commutation relations
of the oscillators. Therefore, we obtain some conditions on the Grassmannian
matrices that appear in these transformations. We observe that there are
several methods to obtain these conditions. In addition, adjoints of the matrix
elements and $T$-duality of these matrices will be obtained. The effects of
this bosonization and fermionization on the mass operators and on some massless
states will be studied. The covariant formalism will be used and hence we
consider both the matter parts and the ghost parts of the superstring theory.
