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We analyse the optimal paths of abatement and carbon prices under a variety of economic, temperature and damage risks. Carbon prices grow in line with economic growth, but with convex damages and temperature-dependent risks of climatic tipping points grow more quickly and with gradual resolution of uncertainty grow more slowly. With temperature-dependent economic damage tipping points carbon prices are higher, but when the tipping point occurs, the price jumps downward. With a temperature cap the efficient carbon price rises at the risk-adjusted interest rate. Allowing for damages as well as a cap leads to a higher carbon price which grows more slowly. But as temperature and cumulative emissions approach their caps, the carbon price is ramped up ever more. Policy makers should expect a rising path of carbon prices.