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The Paris Agreement has opened debate on whether limiting warming to 1.5 degrees C is compatible with current emission pledges and warming of about 0.9 degrees C from the mid-nineteenth century to the present decade. We show that limiting cumulative post-2015 CO2 emissions to about 200 GtC would limit post-2015 warming to less than 0.6 degrees C in 66% of Earth system model members of the CMIP5 ensemble with no mitigation of other climate drivers, increasing to 240 GtC with ambitious non-CO2 mitigation. We combine a simple climate-carbon-cycle model with estimated ranges for key climate system properties from the IPCC Fifth Assessment Report. Assuming emissions peak and decline to below current levels by 2030, and continue thereafter on a much steeper decline, which would be historically unprecedented but consistent with a standard ambitious mitigation scenario (RCP2.6), results in a likely range of peak warming of 1.2-2.0 degrees C above the mid-nineteenth century. If CO2 emissions are continuously adjusted over time to limit 2100 warming to 1.5 degrees C, with ambitious non-CO2 mitigation, net future cumulative CO2 emissions are unlikely to prove less than 250 GtC and unlikely greater than 540 GtC. Hence, limiting warming to 1.5 degrees C is not yet a geophysical impossibility, but is likely to require delivery on strengthened pledges for 2030 followed by challengingly deep and rapid mitigation. Strengthening near-term emissions reductions would hedge against a high climate response or subsequent reduction rates proving economically, technically or politically unfeasible.