The ice giants Uranus and Neptune have hydrogen-based atmospheres with several constituents that condense in their cold upper atmospheres. A small number of bright cloud systems observed in both planets are good candidates for moist convective storms, but their observed properties (size, temporal scales and cycles of activity) differ from moist convective storms in the gas giants. These clouds and storms are possibly due to methane condensation and observations also suggest deeper clouds of hydrogen sulfide (H 2 S) at depths of a few bars. Even deeper, thermochemical models predict clouds of ammonia hydrosulfide (NH 4 SH) and water at pressures of tens to hundreds of bars, forming extended deep weather layers. Because of hydrogen’s low molecular weight and the high abundance of volatiles, their condensation imposes a strongly stabilizing vertical gradient of molecular weight larger than the equivalent one in Jupiter and Saturn. The resulting inhibition of vertical motions should lead to a moist convective regime that differs significantly from the one occurring on nitrogen-based atmospheres like those of Earth or Titan. As a consequence, the thermal structure of the deep atmospheres of Uranus and Neptune is not well understood. Similar processes might occur at the deep water cloud of Jupiter in Saturn, but the ice giants offer the possibility to study these physical aspects in the upper methane cloud layer. A combination of orbital and in situ data will be required to understand convection and its role in atmospheric dynamics in the ice giants, and by extension, in hydrogen atmospheres including Jupiter, Saturn and giant exoplanets. This article is part of a discussion meeting issue ‘Future exploration of ice giant systems’.