  We study the formation of molecular hydrogen on dust grain surfaces and apply
our results to the high redshift universe. We find that a range of physical
parameters, in particular dust temperature and gas temperature, but not so much
dust surface composition, influence the formation rate of H$_2$. The H$_2$
formation rate is found to be suppressed above gas kinetic temperatures of a
few hundred K and for dust temperatures above 200-300 K and below 10 K. We
highlight the differences between our treatment of the H$_2$ formation process
and other descriptions in the literature. We also study the relative importance
of H$_2$ formation on dust grains with respect to molecular hydrogen formation
in the gas phase, through the H$^-$ route. The ratio of formation rates of
these two routes depends to a large part on the dust abundance, on the electron
abundance, and also on the relative strength of the FUV (extra-)galactic
radiation field. We find that for a cosmological evolution of the star
formation rate and dust density consistent with the Madau plot, a positive
feedback effect on the abundance of H$_2$ due to the presence of dust grains
can occur at redshifts $z\ge 3$. This effect occurs for a dust-to-gas mass
ratio as small as 10$^{-3}$ of the galactic value.
