Among renewable energies, wind and solar are inherently intermittent and therefore both require efficient energy storage systems to facilitate a round-the-clock electricity production at a global scale. In this context, concentrated solar power (CSP) stands out among other sustainable technologies because it offers the interesting possibility of storing energy collected from the sun as heat by sensible, latent, or thermochemical means. Accordingly, continuous electricity generation in the power block is possible even during off-sun periods, providing CSP plants with a remarkable dispatchability. Sensible heat storage has been already incorporated to commercial CSP plants. However, because of its potentially higher energy storage density, thermochemical heat storage (TCS) systems emerge as an attractive alternative for the design of next-generation power plants, which are expected to operate at higher temperatures. Through these systems, thermal energy is used to drive endothermic chemical reactions, which can subsequently release the stored energy when needed through a reversible exothermic step. This review analyzes the status of this prominent energy storage technology, its major challenges, and future perspectives, covering in detail the numerous strategies proposed for the improvement of materials and thermochemical reactors. Thermodynamic calculations allow selecting high energy density systems, but experimental findings indicate that sufficiently rapid kinetics and long-term stability trough continuous cycles of chemical transformation are also necessary for practical implementation. In addition, selecting easy-to-handle materials with reduced cost and limited toxicity is crucial for large-scale deployment of this technology. In this work, the possible utilization of materials as diverse as metal hydrides, hydroxides, or carbonates for thermochemical storage is discussed. Furthermore, special attention is paid to the development of redox metal oxides, such as Co3O4/CoO, Mn2O3/Mn3O4, and perovskites of different compositions, as an auspicious new class of TCS materials due to the advantage of working with atmospheric air as reactant, avoiding the need of gas storage tanks. Current knowledge about the structural, morphological, and chemical modifications of these solids, either caused during redox transformations or induced wittingly as a way to improve their properties, is revised in detail. In addition, the design of new reactor concepts proposed for the most efficient use of TCS in concentrated solar facilities is also critically considered. Finally, strategies for the harmonic integration of these units in functioning solar power plants as well as the economic aspects are also briefly assessed.