Liquid crystal forming monomers, typically rods, can be polymerized to form long mesogenic molecules. In contrast to simple rods, these polymers often have internal degrees of freedom so that they display the subtle behaviour of both high polymers and simple liquid crystals. They can have the rod elements either concatenated as a back-bone to give main chain (MC), or pendant to a back-bone to give side chain (SC) liquid crystals, or both. The physics unique to liquid crystalline polymers (LCPS) comes from their shape being dependent on the state of nematic order. Simple systems remain molecular rods (or disks) on ordering whereas a chain extends or flattens (depending on whether or not the nematic order is prolate or oblate). New phenomena as a result of this occur in situations as disparate as networks and, it is predicted, in dielectric response. We examine both SC and MC LCPS and the mechanisms by which they order lyotropically (in solution) and thermotropically (in the melt). Various types of models will be discussed in general and then restricted to the thermotropic case, lyotropic systems being discussed in Lekkerkerker & Vroege (this volume). The transition to the ordered state is first order as in simple nematics. The main characteristics of this state are modified chain conformations and, additionally for side chain polymers, transitions between various novel competing nematic states. A form of self-assembly that is a delicate function of the nematic order is observed in transesterifying LCPS. The number of chain ends is conserved but material exchanged between chains according to whether they are in the isotropic or nematic state. We review a model of this type of self-assembly.