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String theory is currently the most promising candidate for a unifying framework to describe the standard model of particle physics as well as general relativity. One of the areas where most progress has been achieved during the last few years is the AdS/CFT correspondence. In its simplest incarnation it relates string theory on AdS5xS5 to a supersymmetric quantum field theory in four dimensions. The correspondence has helped to shed light on the structure of N = 4 super Yang-Mills theory, and it has led to interesting applications for plasma physics and in condensed matter theory. However, one of the key obstacles towards further progress is that a tractable world-sheet description for strings on AdS5xS5 is so far lacking. There are various ideas for how to make progress with this problem, in particular the pure spinor formulation of Berkovits, as well as the reconstruction programme of Gopakumar. I plan to pursue both of these approaches. In a somewhat separate development, Witten has recently proposed that the AdS/CFT dual of pure gravity in AdS3 is described by an extremal self-dual conformal field theory. A related proposal was subsequently put forward by Strominger and coworkers. In either case, depending on the value of the cosmological constant of the AdS space, the conformal field theories have central charge c = 24k, where k = 1, 2, 3, ... The first theory at c = 24 (k = 1) is the famous Monster conformal field theory, but the theories for k > 1 have so far not been constructed. In the last two years I have given various arguments that suggests that these `extremal' theories are inconsistent for large k. However, the issue has not yet been completely settled, and I would like to use the analysis of higher genus amplitudes to study this problem further.The world-sheet of the string is described by a two-dimensional conformal field theory. Such theories also play an important role for the description of critical phenomena in two dimensions. One class of conformal field theories that have recently attracted a lot of attention are the so-called logarithmic conformal field theories. They appear in various models of statistical physics, for example in the theory of polymers, percolation and various disordered models. They probably also play an important role in Strominger's modification of the pure gravity proposal of Witten. Recently, significant progress (partially by myself) has been made in the general study of logarithmic conformal field theories, but much more remains to be understood. Among other things I want to study logarithmic conformal field theories at c = 0 that are believed to be important for critical systems with quenched disorder.