This review focuses on state-of-the-art research and development in the areas of flexible and stretchable inorganic solar cells, explains the principles behind the main technologies, highlights their key applications, and discusses future challenges. Flexible and stretchable solar cells have gained a growing attention in the last decade due to their ever-expanding range of applications from foldable electronics and robotics to wearables, transportation, and buildings. In this review, we discuss the different absorber and substrate materials in addition to the techniques that have been developed to achieve conformal and elastic inorganic solar cells which show improved efficiencies and enhanced reliabilities compared with their organic counterparts. The reviewed absorber materials range from thin films, including a-Si, copper indium gallium selenide, cadmium telluride, SiGe/III–V, and inorganic perovskite to low-dimensional and bulk materials. The development techniques are generally based on either the transfer-printing of thin cells onto various flexible substrates (e.g., metal foils, polymers, and thin glass) with or without shape engineering, the direct deposition of thin films on flexible substrates, or the etch-based corrugation technique applied on originally rigid cells. The advantages and disadvantages of each of these approaches are analyzed in terms of achieved efficiency, thermal and mechanical reliability, flexibility/stretchability, and economical sustainability.