Photovoltaics (PV) has recently undergone impressive growth and substantial cost decreases, while deployment for concentrating solar power (CSP) has been much slower. As the share of PV rises, the challenge of system integration will increase. This favors CSP, which can be combined with thermal storage and co-firing to reduce variability. It is thus an open question how important solar power will be for achieving climate mitigation targets, and which solar technology will be dominant in the long-term. We address these questions with the state-of-the-art integrated energy-economy-climate model REMIND 1.5, which embodies an advanced representation of the most important drivers of solar deployment. We derive up-to-date values for current and future costs of solar technologies. We calculate a consistent global resource potential dataset for both CSP and PV, aggregated to country-level. We also present a simplified representation of system integration costs of variable renewable energies, suitable for large-scale energy-economy-models. Finally, we calculate a large number of scenarios and perform a sensitivity study to analyze how robust our results are towards future cost reductions of PV and CSP.