Sorption is used for absorption/adsorption heat pumps (sorption refrigeration) and sorption for thermal energy storage (TES). This paper is the first review where the research on both applications is shown together. Sorption has advanced very much due to the immense amount of research carried out around heat pumping and solar refrigeration. Moreover, sorption and thermochemical heat storage attracted considerable attention recently since this technology offers various opportunities in the design of renewable and sustainable energy systems. The paper presents the operation principle of the technology and the materials used or in research are listed and compared. Absorption heat pumping and refrigeration research is today more focussed in the decrease of unit costs and increase of energy efficiency, adsorption is focussed in finding more efficient working pairs, and storage is testing the first prototypes and designing new ones with different or enhanced storage materials and new reactor concepts to optimize energy output. The work is partially funded by the Spanish government (ENE2015-64117-C5-1-R (MINECO/FEDER)). The authors would like to thank the Catalan Government for the quality accreditation given to their research group GREA and DIOPMA (2014 SGR 123, and 2014 SGR 1543, respectively). Dr. Camila Barreneche would like to thank Ministerio de Economia y Competitividad de España for the grant Juan de la Cierva, FJCI-2014-22886.