In this review, we summarise the current status and new trends in concentrating solar power (CSP) technology, analysing the technology cost and their evolution during the last years, with special focus on thermal storage. Moreover, we have carried out a comprehensive review of the molten salts used and proposed in CSP commercial plants. Nitrates, nitrites, chlorides, and carbonates are presented, including their corrosion aspects with common alloys as well as the different possibilities available in the literature to replace them (ternary and quaternary nitrate molten salt) or improve them (addition of nanoparticles). Finally, we have proposed the key factors for a successful new generation of CSP plants, with special focus on high-temperature molten salts (carbonates and chloride blends) and the main important materials requirements for CSP system components. The research leading to these results has received funding from the Spanish government (ENE2015-64117-C5-1-R (MINECO/FEDER)). The authors thank the Catalan Government for the quality accreditation given to their research group GREiA (2017 SGR 1537). GREiA is a certified agent TECNIO in the category of technology developers from the Government of Catalonia. Angel G. Fernández acknowledges the financial support provided by CONICYT/FONDAP 15110019 “Solar Energy Research Center” SERC-Chile. Dr. Aran Solé thanks Ministerio de Economía y Competitividad de España for Grant Juan de la Cierva, FJCI-2015-25741. Dr. Eduard Oró thanks Ministerio de Economia y Competitividad de España for Grant Juan de la Cierva IJCI-2016-30568. This work was authored in part by the National Renewable Energy Laboratory, operated by Alliance for Sustainable Energy, LLC, for the U.S. Department of Energy (DOE) under Contract No. DE-AC36-08GO28308. Funding provided by U.S. Department of Energy, Office of Energy Efficiency and Renewable Energy, Solar Energy Technologies Office, Concentrating Solar Power. The views expressed in the article do not necessarily represent the views of the DOE or the U.S. Government. The U.S. Government retains and the publisher, by accepting the article for publication, acknowledges that the U.S. Government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this work, or allow others to do so, for U.S. Government purposes.