Optical nanofibres are increasingly being used in cold atom experiments due to their versatility and the clear advantages they have when developing all-fibred systems for quantum technologies. They provide researchers with a method of overcoming the Rayleigh range for achieving high intensities in a focussed beam over a relatively long distance, and can act as a noninvasive tool for probing cold atoms. In this review article, we will briefly introduce the theory of mode propagation in an ultrathin optical fibre and highlight some of the more significant theoretical and experimental progresses to date, including the early work on atom probing, manipulation and trapping, the study of atom-dielectric surface interactions, and the more recent observation of nanofibre-mediated nonlinear optics phenomena in atomic media. The functionality of optical nanofibres in relation to the realisation of atom-photon hybrid quantum systems is also becoming more evident as some of the earlier technical challenges are surpassed and, recently, several schemes to implement optical memories have been proposed. We also discuss some possible directions where this research field may head, in particular, in relation to the use of optical nanofibres that can support higher-order modes with an associated orbital angular momentum.