To improve the proficiency of energy systems in addition to increasing the usage of renewable energies, thermal energy storage (TES) is a strategic path. The present literature review reports an overview of the recent advancements in the utilization of salt hydrates (single or binary mixtures) and composites as sorbents for sorption heat storage. Starting by introducing various heat storage systems, the operating concept of the adsorption TES was clarified and contrasted to other technologies. Consequently, a deep examination and crucial problems related to the different types of salt hydrates and adsorbents were performed. Recent advances in the composite materials used in sorption heat storage were also reviewed and compared. A deep discussion related to safety, price, availability, and hydrothermal stability issues is reported. Salt hydrates display high theoretical energy densities, which are promising materials in TES. However, they show a number of drawbacks for use in the basic state including low temperature overhydration and deliquescence (e.g., MgCl2), high temperature degradation, sluggish kinetics leading to a low temperature rise (e.g., MgSO4), corrosiveness and toxicity (e.g., Na2S), and low mass transport due to the material macrostructure. The biggest advantage of adsorption materials is that they are more hydrothermally stable. However, since adsorption is the most common sorption phenomenon, such materials have a lower energy content. Furthermore, when compared to salt hydrates, they have higher prices per mass, which reduces their appeal even further when combined with lower energy densities. Economies of scale and the optimization of manufacturing processes may help cut costs. Among the zeolites, Zeolite 13X is among the most promising. Temperature lifts of 35–45 °C were reached in lab-scale reactors and micro-scale experiments under the device operating settings. Although the key disadvantage is an excessively high desorption temperature, which is problematic to attain using heat sources, for instance, solar thermal collectors. To increase the energy densities and enhance the stability of adsorbents, composite materials have been examined to ameliorate the stability and to achieve suitable energy densities. Based on the reviewed materials, MgSO4 has been identified as the most promising salt; it presents a higher energy density compared to other salts and can be impregnated in a porous matrix to prepare composites in order to overcome the drawbacks connected to its use as pure salt. However, due to pore volume reduction, potential deliquescence and salt leakage from the composite as well as degradation, issues with heat and mass transport can still exist. In addition, to increase the kinetics, stability, and energy density, the use of binary salt deposited in a porous matrix is suitable. Nevertheless, this solution should take into account the deliquescence, safety, and cost of the selected salts. Therefore, binary systems can be the solution to design innovative materials with predetermined sorption properties adapted to particular sorption heat storage cycles. Finally, working condition, desorption temperature, material costs, lifetime, and reparation, among others, are the essential point for commercial competitiveness. High material costs and desorption temperatures, combined with lower energy densities under normal device operating conditions, decrease their market attractiveness. As a result, the introduction of performance metrics within the scientific community and the use of economic features on a material scale are suggested.