The Atlantic Meridional Overturning Circulation has a major impact on climate, not just in the northern Atlantic but globally. Paleoclimatic data show it has been unstable in the past, leading to some of the most dramatic and abrupt climate shifts known. These instabilities are due to two different types of tipping points, one linked to amplifying feedbacks in the large-scale salt transport and the other in the convective mixing that drives the flow. These tipping points present a major risk of abrupt ocean circulation and climate shifts as we push our planet further out of the stable Holocene climate into uncharted waters.