  We investigate a theory in which fundamental objects are branes described in
terms of higher grade coordinates X^{\mu_1 ... \mu_n} encoding both the motion
of a brane as a whole, and its volume evolution. We thus formulate a dynamics
which generalizes the dynamics of the usual branes. Geometrically, coordinates
X^{\mu_1 ... \mu_n} and associated coordinate frame fields {\gamma_{\mu_1 ...
\mu_n}} extend the notion of geometry from spacetime to that of an enlarged
space, called Clifford space or C-space. If we start from 4-dimensional
spacetime, then the dimension of C-space is 16. The fact that C-space has more
than four dimensions suggests that it could serve as a realization of
Kaluza-Klein idea. The "extra dimensions" are not just the ordinary extra
dimensions, they are related to the volume degrees of freedom, therefore they
are physical, and need not be compactified. Gauge fields are due to the metric
of Clifford space. It turns out that amongst the latter gauge fields there also
exist higher grade, antisymmetric fields of the Kalb-Ramond type, and their
non-Abelian generalization. All those fields are naturally coupled to the
generalized branes, whose dynamics is given by a generalized Howe-Tucker action
in curved C-space.
