The panorama after the first superstring revolution consisted of five perturbatively consistent superstring theories, living in a ten-dimensional spacetime. The primary tool for the analysis of perturbative string theory is two-dimensional conformal field theory. This is a very rich algebro-geometric theory with connections to a wide range of mathematical topics like representation theory of infinite-dimensional Lie algebras and algebraic geometry, and also to physical topics like statistical mechanics and condensed matter. The need to relate the ten-dimensional string theory to the four-dimensional physics we observe initiated an intensive study of certain six-dimensional complex geometries known as Calabi-Yau manifolds. A consequence of these investigations is the mirror symmetry conjecture, which has occupied a growing number of mathematicians ever since.

As a result of the second revolution the picture has now been drastically altered. We now understand that the five ten-dimensional theories, and a newly discovered eleven-dimensional theory, are but different manifestations of an underlying theory (code-named M-theory). Moreover the different 'subtheories' are related by duality transformations, akin to electromagnetic duality but generalising it in nontrivial ways. A suggestive analogy is that of the description of a manifold in terms of coordinate charts and transition functions. Here the different subtheories play the role of the coordinate charts and the duality transformations play the role of the transition functions. Continuing with this analogy, we know that in the case of manifolds it is possible to define them without having to resort to local charts; and the search for such an intrinsic description of M-theory is one of the main problems in the field today and one to which a lot of effort continues to be devoted.

Instrumental in the second superstring revolution are the so-called branes. These appeared initially as classical solutions of the supergravity theories which are the field-theory limits of the string theories and of the new eleven-dimensional theory. Some of these branes, the so-called D-branes, can also be understood as Dirichlet-type boundary conditions for open strings and hence amenable to the methods of boundary conformal field theory.

D-branes have also been instrumental in the so-called gauge/gravity correspondence. Initially conjectured by Maldacena in 1997, this correspondence states that there is a weak/strong coupling duality between (the 't Hooft limit of) supersymmetric gauge theory and (the supergravity limit of) superstring theory. This correspondence hints that string theory and gauge theory are but two sides of the same coin and not the drastically different theories that they appear to be.

Much effort continues to be devoted to testing and exploiting this correspondence. Recently a novel large rank limit of the gauge theory, inspired by Penrose's plane-wave limit in gravity, allows for the first time a perturbative test of the correspondence simultaneously in the gravity and gauge theory sides, albeit for a special sub-sector of the theory.