  We provide a classification of the IIB D$p$- and NS$p$-branes in which the
brane action exists due to a non-trivial class of the Chevalley-Eilenberg
cohomology of free differential algebras. We then present a new geometric
formulation of the IIB D$p$- and NS$p$-branes ($p\leq 5$) in which the
manifestly superinvariant Wess-Zumino terms are constructed in terms of the
supersymmetric currents. The supercurrents are obtained by using supergroup
manifolds corresponding to the IIB-brane superalgebra, which is characterized
by the generators of D3-, D5-, NS5- and KK5-branes in addition to the
previously introduced generators of supertranslations, F- and D-strings. The
charges of D1-, F1- and D3-branes are related to those of the M-algebra, but
some charges of D5- and NS5-branes are not. The S-duality of the type-IIB
theory is realized as transformations of the supercurrents generalizing the
SO(2) R-symmetry of the superalgebra. We thus find that the superalgebra is
lifted into twelve-dimensions with signature (11,1).
