We develop mathematical tools to study critical processes on planet earth due to climate change. Climate tipping points are an example of such critical processes and they occur when change in a part of the climate system becomes self-perpetuating beyond a warming threshold, leading to substantial earth system impacts. We are particularly interested in tipping points in the earth's cryosphere (sea ice loss, Arctic phytoplankton bloom, permafrost-induced methane emission). We also develop the mathematical theory of mass extinction events.