  We show that supersymmetric and $\kappa$-symmetric Dp-brane actions in type
II supergravity background have the same duality transformation properties as
those in a flat Minkowskian background. Specially, it is shown that the super
D-string transforms in a covariant way while the super D3-brane is self-dual
under the $SL(2,Z)$ duality. Also, the D2-brane and the D4-brane transform in
ways expected from the relation between type IIA superstring theory and
M-theory. The present study proves that various duality symmetries, which were
originally found in the flat background field, are precisely valid even in the
curved background geometry.
