Conventional solid-core optical fibres require highly transparent materials. Such materials have been difficult to identify owing to the fundamental limitations associated with the propagation of light through solids, such as absorption, scattering and nonlinear effects. Hollow optical fibres offer the potential to minimize the dependence of light transmission on fibre material transparency. Here we report on the design and drawing of a hollow optical fibre lined with an interior omnidirectional dielectric mirror. Confinement of light in the hollow core is provided by the large photonic bandgaps established by the multiple alternating submicrometre-thick layers of a high-refractive-index glass and a low-refractive-index polymer. The fundamental and high-order transmission windows are determined by the layer dimensions and can be scaled from 0.75 to 10.6 micro m in wavelength. Tens of metres of hollow photonic bandgap fibres for transmission of carbon dioxide laser light at 10.6 micro m wavelength were drawn. The transmission losses are found to be less than 1.0 dB m(-1), orders of magnitude lower than those of the intrinsic fibre material, thus demonstrating that low attenuation can be achieved through structural design rather than high-transparency material selection.