Liquid crystals are complex materials that share properties of both solids and liquids. This is a consequence of complex anisotropic molecules that permit establishing phases with orientational and positional orders. Thus, a large variety of phases and phase transitions can occur in these systems. After a detailed description of general features of these materials, the tensorial nature of the orientational order parameter is discussed. Then, the Landau–de Gennes theory is developed for the isotropic–nematic transition. Later, positional degrees of freedom are included to account for the nematic–smectic transition. Next, the theory is generalized to include fluctuations, distortions and the effect of an external field. In the last part, topological defects are discussed with a particular emphasis on defects such as skyrmions and merons which can form in chiral liquid crystals such as cholesteric and blue phases. Finally, the analogy of these classes of defects with those occurring in non-collinear magnetic materials is considered.

This chapter treats incompressible flocks in two dimensions, and shows that they map onto both equilibrium two-dimensional smectics, and our old friend the KPZ equation (albeit in one dimension), as well as a peculiar type of constrained magnet. Exact scaling laws are again found, this time by exploiting these mappings.

Off-lattice models both based on purely repulsive or attractive-repulsive Gay–Berne models allow us to simulate liquid crystal phases with some positional as well as orientational order. This chapter summarizes simulation results for anisotropic particles of elongated or discotic shape of the two types either pristine or decorated with charges, dipoles and quadrupoles. Beyond showing the effect of key molecular features (e.g. aspect ratios) on morphologies and phase diagrams, applications specific to liquid crystals, like the calculation of elastic constants and the simulation of a TN LCD, are reported. Tapered, bowlic and biaxial GB type single particle systems as well as more complex ones based of multi-particle mesogens (banana phases, polymers, elastomers) are discussed.