  A new family of supergravity theories in odd dimensions is presented. The
Lagrangian densities are Chern-Simons forms for the connection of a
supersymmetric extension of the anti-de Sitter algebra. The superalgebras are
the supersymmetric extensions of the AdS algebra for each dimension, thus
completing the analysis of van Holten and Van Proeyen, which was valid for N=1
and for D=2,3,4,mod 8. The Chern-Simons form of the Lagrangian ensures
invariance under the gauge supergroup by construction and, in particular, under
local supersymmetry. Thus, unlike standard supergravity, the local
supersymmetry algebra closes off-shell and without requiring auxiliary fields.
The Lagrangian is explicitly given for D=5, 7 and 11. In all cases the
dynamical field content includes the vielbein, the spin connection, N
gravitini, and some extra bosonic ``matter'' fields which vary from one
dimension to another. The superalgebras fall into three families: osp(m|N) for
D=2,3,4, mod 8, osp(N|m) for D=6,7,8, mod 8, and su(m-2,2|N) for D=5 mod 4,
with m=2^{[D/2]}. The possible connection between the D=11 case and M-Theory is
also discussed.
