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This paper presents a mixed integer, multi-period, cost-minimising model for a carbon capture, transport and storage (CCTS) network in Europe. The model incorporates endogenous decisions about carbon capture, pipeline and storage investments. The capture, flow and injection quantities are based on given costs, certificate prices, storage capacities and point source emissions. The results indicate that CCTS can theoretically contribute to the decarbonisation of Europe’s energy and industrial sectors. This requires a CO2 certificate price rising to €55 per tCO2 in 2050, and sufficient CO2 storage capacity available for both on- and offshore sites. Yet CCTS deployment is highest in CO2-intensive industries where emissions cannot be avoided by fuel switching or alternative production processes. In all scenarios, the importance of the industrial sector as a first-mover to induce the deployment of CCTS is highlighted. By contrast, a decrease in available storage capacity or a more moderate increase in CO2 prices will significantly reduce the role of CCTS as a CO2 mitigation technology, especially in the energy sector. Furthermore, continued public resistance to onshore CO2 storage can only be overcome by constructing expensive offshore storage. Under this restriction, reaching the same levels of CCTS penetration would require a doubling of CO2 certificate prices.