Ambitious targets for renewable energy and CO2 taxation both represent political instruments for decarbonisation of the energy system. We model a high number of coupled electricity and heating systems, where the primary sources of CO2 neutral energy are from variable renewable energy sources (VRES), Le., wind and solar generators. The model includes hourly dispatch of all technologies for a full year for every country in Europe. In each model run, the amount of renewable energy and the level of CO2 tax are fixed exogenously, while the cost-optimal composition of energy generation, conversion, transmission and storage technologies and the corresponding CO2 emissions are calculated. We show that even for high penetrations of VRES, a significant CO2 tax of more than 100 (sic)/CO2 is required to limit the combined CO2 emissions from the sectors to less than 5% of 1990 levels, because curtailment of VRES, combustion of fossil fuels and inefficient conversion technologies are economically favoured despite the presence of abundant VRES. A sufficiently high CO2 tax results in the more efficient use of VRES by means of heat pumps and hot water storage, in particular. We conclude that a renewable energy target on its own is not sufficient; in addition, a CO2 tax is required to decarbonise the electricity and heating sectors and incentivise the least cost combination of flexible and efficient energy conversion and storage.