  The forms in D-dimensional (half-)maximal supergravity theories are discussed
for 3 $\leq$ D $\leq$ 11. Superspace methods are used to derive consistent sets
of Bianchi identities for all the forms for all degrees, and to show that they
are soluble and fully compatible with supersymmetry. The Bianchi identities
determine Lie superalgebras that can be extended to Borcherds superalgebras of
a special type. It is shown that any Borcherds superalgebra of this type gives
the same form spectrum, up to an arbitrary degree, as an associated Kac-Moody
algebra. For maximal supergravity up to D-form potentials, this is the very
extended Kac-Moody algebra E11. It is also shown how gauging can be carried out
in a simple fashion by deforming the Bianchi identities by means of a new
algebraic element related to the embedding tensor. In this case the appropriate
extension of the form algebra is a truncated version of the so-called tensor
hierarchy algebra.
