A survey of recent developments in the field of non-linear analysis and the geometry of mappings. Sobolev mappings, quasiconformal mappings, or deformations, between subsets of Euclidean space, or manifolds or more general geometric objects may arise as the solutions to certain optimization problems in the calculus of variations or in non-linear elasticity, as the solutions to differential equations (particularly in conformal geometry), as local co-ordinates on a manifold or as geometric realizations of abstract isomorphisms between spaces such as those that arise in dynamical systems (for...

The 'measurable Riemann Mapping Theorem' has found a central role in a diverse variety of areas such as holomorphic dynamics, Teichmuller theory, low dimensional topology and geometry, and the planar theory of PDEs. The authors recount aspects of this clas