The effective action for type II string theory compactified on a six torus is N = 8 supergravity, which is known to have an E 7 duality symmetry. We show that this is broken by quantum effects to a discrete subgroup, E 7 (Z), which contains both the T-duality group O(6, 6; Z) and the S-duality group SL(2; Z). We present evidence for the conjecture that E 7 (Z) is an exact 'U-duality' symmetry of type II string theory. This conjecture requires certain extreme black hole states to be identified with massive modes of the fundamental string. The gauge bosons from the Ramond-Ramond sector couple not to string excitations but to solitons. We discuss similar issues in the context of toroidal string compactifications to other dimensions, compactifications of the type II string on K 3 ×T 2 and compactifications of eleven-dimensional supermembrane theory.

All purely bosonic supersymmetric solutions of minimal supergravity in five dimensions are classified. The solutions preserve either one half or all of the supersymmetry. Explicit examples of new solutions are given, including a large family of plane-fronted waves and a maximally supersymmetric analogue of the Gödel universe which lifts to a solution of eleven dimensional supergravity that preserves 20 supersymmetries.

A geometric string solution has background fields in overlapping coordinate patches related by diffeomorphisms and gauge transformations, while for a non-geometric background this is generalised to allow transition functions involving duality transformations. Non-geometric string backgrounds arise from T-duals and mirrors of flux compactifications, from reductions with duality twists and from asymmetric orbifolds. Strings in 'T-fold' backgrounds with a local n-torus fibration and T-duality transition functions in O(n, n; ) are formulated in an enlarged space with a T 2n fibration which is geometric, with spacetime emerging locally from a choice of a T n submanifold of each T 2n fibre, so that it is a subspace or brane embedded in the enlarged space. T-duality acts by changing to a different T n subspace of T 2n . For a geometric background, the local choices of T n fit together to give a spacetime which is a T n bundle, while for non-geometric string backgrounds they do not fit together to form a manifold. In such cases spacetime geometry only makes sense locally, and the global structure involves the doubled geometry. For open strings, generalised D-branes wrap a T n subspace of each T 2n fibre and the physical D-brane is the part of the physical space lying in the generalised D-brane subspace.

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The zero modes of closed strings on a torus -the torus coordinates plus dual coordinates conjugate to winding number-parameterize a doubled torus. In closed string field theory, the string field depends on all zero-modes and so can be expanded to give an infinite set of fields on the doubled torus. We use string field theory to construct a theory of massless fields on the doubled torus. Key to the consistency is a constraint on fields and gauge parameters that arises from the L 0 −L 0 = 0 condition in closed string theory. The symmetry of this double field theory includes usual and 'dual diffeomorphisms', together with a T-duality acting on fields that have explicit dependence on the torus coordinates and the dual coordinates. We find that, along with gravity, a Kalb-Ramond field and a dilaton must be added to support both usual and dual diffeomorphisms. We construct a fully consistent and gauge invariant action on the doubled torus to cubic order in the fields. We discuss the challenges involved in the construction of the full nonlinear theory. We emphasize that the doubled geometry is physical and the dual dimensions should not be viewed as an auxiliary structure or a gauge artifact.