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A new family of supergravity theories in odd dimensions is presented. TheLagrangian densities are Chern-Simons forms for the connection of asupersymmetric extension of the anti-de Sitter algebra. The superalgebras arethe supersymmetric extensions of the AdS algebra for each dimension, thuscompleting the analysis of van Holten and Van Proeyen, which was valid for N=1and for D=2,3,4,mod 8. The Chern-Simons form of the Lagrangian ensuresinvariance under the gauge supergroup by construction and, in particular, underlocal supersymmetry. Thus, unlike standard supergravity, the localsupersymmetry algebra closes off-shell and without requiring auxiliary fields.The Lagrangian is explicitly given for D=5, 7 and 11. In all cases thedynamical field content includes the vielbein, the spin connection, Ngravitini, and some extra bosonic ``matter'' fields which vary from onedimension to another. The superalgebras fall into three families: osp(m|N) forD=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 isalso discussed.Comment: 13pages, RevTeX, no figures, two column

CHERN-SIMONS SUPERGRAVITIES WITH OFF-SHELL LOCAL SUPERALGEBRAS