Low-frequency radio observations of neutral hydrogen during and before the epoch of cosmic re-ionization will provide ∼1000 quasi-independent source planes, each of precisely known redshift, if a resolution of ∼1 arcmin or better can be attained. These planes can be used to reconstruct the projected mass distribution of foreground material. Structure in these source planes is linear and Gaussian at high redshift (30 < z < 300) but is non-linear and non-Gaussian during re-ionization. At both epochs, significant power is expected down to subarcsecond scales. We demonstrate that this structure can, in principle, be used to make mass images with a formal signal-to-noise ratio (S/N) per pixel exceeding 10, even for pixels as small as an arcsecond. With an ideal telescope, both resolution and S/N can exceed those of even the most optimistic idealized mass maps from galaxy lensing by more than an order of magnitude. Individual dark haloes similar in mass to that of the Milky Way could be imaged with high S/N out to z∼ 10. Even with a much less ambitious telescope, a wide-area survey of 21-cm lensing would provide very sensitive constraints on cosmological parameters, in particular on dark energy. These are up to 20 times tighter than the constraints obtainable from comparably sized, very deep surveys of galaxy lensing, although the best constraints come from combining data of the two types. Any radio telescope capable of mapping the 21-cm brightness temperature with good frequency resolution (∼0.05 MHz) over a band of width ≳10 MHz should be able to make mass maps of high quality. The planned Square Kilometre Array may be able to map the mass with moderate S/N down to arcminute scales, depending on the re-ionization history of the universe and the ability to subtract foreground sources.