The increasing demands for cooling/heating, depletion of fossil fuels, and greenhouse gases emissions promote the development of adsorption heat transformation and storage (AHTS). This emerging technology is especially promising for converting low-temperature heat, like environmental, solar, and waste heat. Among the known AHTS applications (cooling, heat pumping, amplification, and storage), the adsorption heat storage and amplification are less developed, thus gaining an increasing attention of the scientific community. The researchers are mainly focused on the developing new cycles for heat storage/amplification and advanced adsorbents specialized for these cycles. In this paper, we review the state-of-the-art in the fields of adsorption heat storage/amplification. The new, recently suggested, cycles (e.g. a “Heat from Cold” cycle for upgrading the ambient heat) will be described and analyzed from both thermodynamic and dynamic points of view. New adsorbents developed for adsorption heat storage/amplification will be presented. Special attention will be paid to the problem how to harmonize the adsorbent with the AHTS cycle under various climatic conditions. The lab-scale units constructed for verification of the cycle feasibility and adsorbent efficiency also are briefly described and analyzed. Finally, the problems and outlooks of adsorption heat storage/amplification will be discussed.