Microporous carbons are important in a wide variety of applications, ranging from pollution control to supercapacitors, yet their structure at the molecular level is poorly understood. Over the years, many structural models have been put forward, but none has been entirely satisfactory in explaining the properties of the carbons. The discovery of fullerenes and fullerene-related structures such as carbon nanotubes gave us a new perspective on the structure of solid carbon, and in 1997 it was suggested that microporous carbon may have a structure related to that of the fullerenes. Recently, evidence in support of such a structure has been obtained using aberration-corrected transmission electron microscopy, electron energy loss spectroscopy and other techniques. This article describes the development of ideas about the structure of microporous carbon, and reviews the experimental evidence for a fullerene-related structure. Theoretical models of the structural evolution of microporous carbon are summarised, and the use of fullerene-like models to predict the adsorptive properties of microporous carbons are reviewed.