Recent years have seen intense interaction between mathematics and materials science, including solid mechanics and liquid crystals. This has borne much fruit in solid mechanics, the explanation of intriguing material behaviour (e.g., the shape memory effects) by mathematical models that relate behaviour to microstructure; in liquid crystals, models relating static and dynamic properties to the existence and regularity of harmonic maps, possibly with defects, between topologically nontrivial spaces. This successful interaction has in turn raised a number of questions, many of which are of interest simultaneously in mathematics, in the physical sciences and in engineering. The relevant mathematical areas are primarily, but not exclusively, calculus of variations, partial differential equations, functional and real analysis and topology. Specific research problems include the following. Solid mechanics: microstructure evolution in solids, homogenization of (polycrystalline) materials with degenerate energy, computation of quasiconvex hulls and morphology formation in biological tissues as a result of stresses induced by growth. Liquid crystals: topological classification, energy bounds, for nematics in polyhedral geometries with natural, e.g. tangent, boundary conditions; number of smooth solutions, regularity of weak solutions; switching mechanisms, applications to bistable display technology.