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We present a tentative formulation of theories of gravity with suitablematter content, including in particular pure gravity in D dimensions, thebosonic effective actions of M-theory and of the bosonic string, in terms ofactions invariant under very-extended Kac-Moody algebras G+++. We conjecturethat they host additional degrees of freedom not contained in the conventionaltheories. The actions are constructed in a recursive way from a level expansionfor all very-extended algebras G+++. They constitute non-linear realisations oncosets, a priori unrelated to space-time, obtained from a modified Chevalleyinvolution. Exact solutions are found for all G+++. They describe the algebraicproperties of BPS extremal branes, Kaluza-Klein waves and Kaluza-Kleinmonopoles. They illustrate the generalisation to all G+++ invariant theories ofthe well-known duality properties of string theories by expressing duality asWeyl invariance in G+++. Space-time is expected to be generated dynamically. Inthe level decomposition of E8+++ = E11, one may indeed select an A10representation of generators Pa which appears to engender space-timetranslations by inducing infinite towers of fields interpretable as fieldderivatives in space and time.Comment: Latex 45 pages, 1 figure. Discussion on pages 19 and 20 altered.  Appendix B amplified. 4 footnotes added. 2 references added. Acknowledgments  updated. Additional minor correction

G+++ Invariant Formulation of Gravity and M-Theories: Exact BPS Solutions