Climate control applications in the form of heating and cooling account for a significant portion of energy consumption in buildings and transportation. Consequently, improved efficiency of climate control systems can significantly reduce the energy consumption and greenhouse gas emissions. In particular, by leveraging intermittent or continuous sources of waste heat and solar energy, thermally-driven energy storage systems for climate control can play a crucial role. We demonstrate the concept of a thermophysical battery, which operates by storing thermal energy and subsequently releasing it to provide heating and cooling on demand. Taking advantage of the adsorption-desorption and evaporation-condensation mechanisms, the thermophysical battery can be a high-power density and rechargeable energy storage system. We investigated the thermophysical battery in detail to identify critical parameters governing its overall performance. A detailed computational analysis was used to predict its cyclic performance when exposed to different operating conditions and thermodynamic cycles. In addition, an experimental test bed was constructed using a contemporary adsorptive material, NaX-zeolite, to demonstrate this concept and deliver average heating and cooling powers of 900W and 650W, respectively. The maximum power densities and specific powers observed were 103W/l and 65W/kg for heating, and 78W/l and 49W/kg for cooling, respectively, making the thermophysical battery competitive with the state-of-the-art climate control systems that provide relatively lower power densities. Additionally, with further opportunities for development and innovation, especially in synthesizing novel adsorptive materials, the thermophysical battery can achieve significantly higher power densities. With its ability to function using thermal energy input while being compact and lightweight, the thermophysical battery offers an option to address the energy challenges associated with the rising demand for climate control.