This workshop took place in the Mathematics Department of Imperial College London from *July 30* to *Aug 8 2008*.

Geometry is one of the most diverse of all areas of mathematics. Following Klein's Erlangen program, geometry defined itself in the twentieth century predominantly as the study of the properties of an object invariant under a group of symmetries, and led to many important results ranging from group theory to differential equations. From a different angle, the work of Coxeter illustrates the beautiful structures that can be revealed from studying simple geometric objects like polytopes. The study of tilings brings together these two themes of geometry.

The central question in the theory of tiling is whether a set of shapes can tile the plane or not. This question goes back to the Ancient Greeks, who were certainly aware that the equilateral triangle, square and regular hexagon are the only regular polygons to tile the plane. Tiling questions were also investigated by Kepler and the artistic work of Islamic artists. Mathematical work concentrated on periodic tilings, but this study was part of the development of group theory. Initially with the space groups, the groups of symmetries of periodic tilings on the plane and in higher dimensions and also in the discovery of sporadic finite simple groups. The study of periodic and symmetric structures continues to be developed. This year sees the publication of Symmetries of Things by Conway, Goodman-Strauss and Burgiel, a comprehensive study of the area which includes a new proof of Conway of the characterisation of space groups on the plane.

The general question emerges in Hilbert's 18th problem, but was first stated explicitly by Hao Wang in 1961 as the Domino Problem: Does an algorithm exist that tells if a given set of tiles will tile the plane. This problem was answered in the negative on the euclidean plane (and thus in all euclidean space of dimension greater than 1, where the problem is trivial) by Berger in 1964. Berger showed the the problem was undecidable. One consequence of this is that there exist sets of shapes that do not tile periodically, called aperiodic protosets. Berger found such an aperiodic protoset, but it contained nearly 20,000 tiles. This number was gradually brought down until the discovery of the famous Penrose tiling with just 2 tiles.

These new results combined with a need for simple models of non-periodic, but ordered, structures from physics (for examples as models for the structure of quasicrystals) have led to a great deal of research in tiling theory. In most cases however the research has been carried out very close to the area of application. The present workshop follows on from a special session at a regional AMS meeting in Davidson, NC, USA in March 2007. That meeting was well attended and generated a lot of interest. The main goal of this workshop is to build on that success: forging stronger links between different researchers working with tiling and communicating the exciting new developments in several areas to other researchers in tilings and its applications.