Translating the unique characteristics of individual single-walled carbon nanotubes into macroscopic materials such as fibres and sheets has been hindered by ineffective assembly. Fluid-phase assembly is particularly attractive, but the ability to dissolve nanotubes in solvents has eluded researchers for over a decade. Here, we show that single-walled nanotubes form true thermodynamic solutions in superacids, and report the full phase diagram, allowing the rational design of fluid-phase assembly processes. Single-walled nanotubes dissolve spontaneously in chlorosulphonic acid at weight concentrations of up to 0.5 wt%, 1,000 times higher than previously reported in other acids. At higher concentrations, they form liquid-crystal phases that can be readily processed into fibres and sheets of controlled morphology. These results lay the foundation for bottom-up assembly of nanotubes and nanorods into functional materials.