  A differential calculus of the first order over multi-braided quantum groups
is developed. In analogy with the standard theory, left/right-covariant and
bicovariant differential structures are introduced and investigated.
Furthermore, antipodally covariant calculi are studied. The concept of the
*-structure on a multi-braided quantum group is formulated, and in particular
the structure of left-covariant *-covariant calculi is analyzed. A special
attention is given to differential calculi covariant with respect to the action
of the associated braid system. In particular it is shown that the left/right
braided-covariance appears as a consequence of the left/right-covariance
relative to the group action. Braided counterparts of all basic results of the
standard theory are found.
