CCS is expected to play a sizeable role over the next several decades to drastically abate greenhouse gas emissions in the energy system by 2050. With the integrated assessment model TIAM-ECN we project how large its contribution could be in Europe in the power sector and industry, and analyze how this contribution may be affected by cost decline (driven by technological innovation) and ambitious greenhouse gas emission reduction targets. If Europe stays on track of achieving zero net emissions by mid-century, we find that CO2 capture activities in Europe may grow to a total amount of some 1700 MtCO2/yr in 2050. We project that by 2050 power plants equipped with CCS could contribute by approximately 25 % to total electricity generation, in competition with low-carbon alternatives such as nuclear energy and renewables like solar photovoltaic and wind power. CCS in association with the use of natural gas could account for about 35 % of energy use in industrial applications, while the use of biomass and coal combined with CCS could be employed for the production of hydrogen. We find that stringent climate policy measures constitute a bigger driver of broad CCS diffusion than CCS technology cost reductions. We project that under stringent climate policy targeting zero net emissions by 2050, a 30 % CCS capital cost reduction as a result of technological advancement could proffer Europe by the middle of the century a saving of about 17 billion US$/yr in additional energy system costs.