We present a general rule determining how extremal branes can interesect in a configuration with zero binding energy. The rule is derived in a model independent way and in arbitrary spacetime dimensions D by solving the equations of motion of gravity coupled to a dilaton and several different n-form field strengths. The intersection rules are all compatible with supersymmetry, although derived without using it. We then specialize to the branes occurring in type II string theories and in M-theory. We show that the intersection rules are consistent with the picture that open branes can have boundaries on some other branes. In particular, all the D-branes of dimension q, with 1 ≤ q ≤ 6, can have boundaries on the solitonic 5-brane.

We consider the Cartan subalgebra of any very extended algebra G +++ where G is a simple Lie algebra and let the parameters be space-time fields. These are identified with diagonal metrics and dilatons. Using the properties of the algebra, we find that for all very extensions G +++ of simple Lie algebras there are theories of gravity and matter, which admit classical solutions carrying representations of the Weyl group of G +++ . We also identify the T and S-dualities of superstrings and of the bosonic string with Weyl reflections and outer automorphisms of well-chosen very extended algebras and we exhibit specific features of the very extensions. We take these results as indication that very extended algebras underlie symmetries of any consistent theory of gravity and matter, and might encode basic information for the construction of such theory.

We present a tentative formulation of theories of gravity with suitable matter content, including in particular pure gravity in D dimensions, the bosonic effective actions of M-theory and of the bosonic string, in terms of actions invariant under very-extended Kac-Moody algebras G +++ . We conjecture that they host additional degrees of freedom not contained in the conventional theories. The actions are constructed in a recursive way from a level expansion for all very-extended algebras G +++ . They constitute non-linear realisations on cosets, a priori unrelated to space-time, obtained from a modified Chevalley involution. Exact solutions are found for all G +++ . They describe the algebraic properties of BPS extremal branes, Kaluza-Klein waves and Kaluza-Klein monopoles. They illustrate the generalisation to all G +++ invariant theories of the well-known duality properties of string theories by expressing duality as Weyl invariance in G +++ . Space-time is expected to be generated dynamically. In the level decomposition of E +++ 8 = E 11 , one may indeed select an A 10 representation of generators P a which appears to engender space-time translations by inducing infinite towers of fields interpretable as field derivatives in space and time.

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