Permafrost can potentially release more than twice as much carbon than is currently in the atmosphere, and is warming at a rate twice as fast as the rest of the planet.  Fundamentally, the thawing permafrost is a phase transition phenomenon, where a solid turns to liquid, albeit on large regional scales and over a period of time that depends on environmental forcing and other factors. In this talk, we present mathematical models that help to understand the processes on the interface “frozen ground-atmosphere” and investigate their criticality.