  We discuss the r\^ole of enlarged superspaces in two seemingly different
contexts, the structure of the $p$-brane actions and that of the
Cremmer-Julia-Scherk eleven-dimensional supergravity. Both provide examples of
a common principle: the existence of an {\it enlarged superspaces
coordinates/fields correspondence} by which all the (worldvolume or spacetime)
fields of the theory are associated to coordinates of enlarged superspaces. In
the context of $p$-branes, enlarged superspaces may be used to construct
manifestly supersymmetry-invariant Wess-Zumino terms and as a way of expressing
the Born-Infeld worldvolume fields of D-branes and the worldvolume M5-brane
two-form in terms of fields associated to the coordinates of these enlarged
superspaces. This is tantamount to saying that the Born-Infeld fields have a
superspace origin, as do the other worldvolume fields, and that they have a
composite structure. In $D$=11 supergravity theory enlarged superspaces arise
when its underlying gauge structure is investigated and, as a result, the
composite nature of the $A_3$ field is revealed: there is a full one-parametric
family of enlarged superspace groups that solve the problem of expressing $A_3$
in terms of spacetime fields associated to their coordinates. The corresponding
enlarged supersymmetry algebras turn out to be deformations of an {\it
expansion} of the $osp(1|32)$ algebra. The unifying mathematical structure
underlying all these facts is the cohomology of the supersymmetry algebras
involved.
