An ideal fibre-reinforced fluid is incompressible and inextensible along a family of material curves that are convected with the fluid. It is a model for continuous fibre-resin systems in the fluid state in which forming processes take place. Like liquid crystals, these fluids have strong directional properties. The kinematic and constitutive theory of ideal fibre-reinforced fluids is described, with particular reference to plane flows. The class of flows in which the fibres are aligned along the streamlines is considered, and an explanation is given for the observed prevalence of this class of flows.