The fundamental structure of matter and spacetime at the shortest length scales remains an exciting frontier of basic research in theoretical physics. A unifying theme in this area is the quantization of geometrical objects. The majority of lectures at the Advanced Study Institute on Quantum Ge ometry in Akureyri was on recent advances in superstring theory, which is the leading candidate for a unified description of all known elementary par ticles and interactions. The geometric concept of one-dimensional extended objects, or strings, has always been at the core of superstring theory but in recent years the focus has shifted to include also higher-dimensional ob jects, so called D-branes, which play a key role in the non-perturbative dynamics of the theory. A related development has seen the strong coupling regime of a given string theory identified with the weak coupling regime of what was previ ously believed to be a different theory, and a web of such" dualities" that interrelates all known superstring theories has emerged. The resulting uni fied theoretical framework, termed M-theory, has evolved at a rapid pace in recent years."

This graduate/research level text describes in a unified fashion the statistical mechanics of random walks, random surfaces and random higher dimensional manifolds with an emphasis on the geometrical aspects of the theory and applications to the quantisation of strings, gravity and topological field theory. With chapters on random walks, random surfaces, two- and higher dimensional quantum gravity, topological quantum field theories and Monte Carlo simulations of random geometries, the text provides a self-contained account of quantum geometry from a statistical field theory point of view. The approach uses discrete approximations and develops analytical and numerical tools. Continuum physics is recovered through scaling limits at phase transition points and the relation to conformal quantum field theories coupled to quantum gravity is described. The most important numerical work is covered, but the main aim is to develop mathematically precise results that have wide applications. Many diagrams and references are included.

The fundamental structure of matter and spacetime at the shortest length scales remains an exciting frontier of basic research in theoretical physics. A unifying theme in this area is the quantization of geometrical objects. The majority of lectures at the Advanced Study Institute on Quantum Ge ometry in Akureyri was on recent advances in superstring theory, which is the leading candidate for a unified description of all known elementary par ticles and interactions. The geometric concept of one-dimensional extended objects, or strings, has always been at the core of superstring theory but in recent years the focus has shifted to include also higher-dimensional ob jects, so called D-branes, which play a key role in the non-perturbative dynamics of the theory. A related development has seen the strong coupling regime of a given string theory identified with the weak coupling regime of what was previ ously believed to be a different theory, and a web of such" dualities" that interrelates all known superstring theories has emerged. The resulting uni fied theoretical framework, termed M-theory, has evolved at a rapid pace in recent years."

This graduate/research level text describes in a unified fashion the statistical mechanics of random walks, random surfaces and random higher dimensional manifolds with an emphasis on the geometrical aspects of the theory and applications to the quantisation of strings, gravity and topological field theory. With chapters on random walks, random surfaces, two- and higher dimensional quantum gravity, topological quantum field theories and Monte Carlo simulations of random geometries, the text provides a self-contained account of quantum geometry from a statistical field theory point of view. The approach uses discrete approximations and develops analytical and numerical tools. Continuum physics is recovered through scaling limits at phase transition points and the relation to conformal quantum field theories coupled to quantum gravity is described. The most important numerical work is covered, but the main aim is to develop mathematically precise results that have wide applications. Many diagrams and references are included.