There is widespread concern that anthropogenic global warming will trigger Arctic climate tipping points. The Arctic has a long history of natural, abrupt climate changes, which together with current observations and model projections, can help us to identify which parts of the Arctic climate system might pass future tipping points. Here the climate tipping points are defined, noting that not all of them involve bifurcations leading to irreversible change. Past abrupt climate changes in the Arctic are briefly reviewed. Then, the current behaviour of a range of Arctic systems is summarised. Looking ahead, a range of potential tipping phenomena are described. This leads to a revised and expanded list of potential Arctic climate tipping elements, whose likelihood is assessed, in terms of how much warming will be required to tip them. Finally, the available responses are considered, especially the prospects for avoiding Arctic climate tipping points.