Snow is the most reflective, and also the most insulative, natural surface on Earth. Consequently, snow cover on land and sea ice is an integral part of the climate system. Recent thinning of snow cover on Arctic sea ice has contrasting effects: reduced insulation from thin snow promotes sea ice formation, while increased absorption of sunlight from thin snow promotes sea ice melt. So, what is a thinner snow cover's net effect on Arctic sea ice in a warmer climate? This talk presents the major challenges we face in observing, understanding and modelling the snow-ice system under anthropogenic warming, and advances stimulated by bringing field observations, remote sensing, and modelling together.