Tunable rigidity materials have potentially widespread implications in robotic technologies. They enable morphological shape change while maintaining structural strength, and can reversibly alternate between rigid, load bearing and compliant, flexible states capable of deformation within unstructured environments. In this review, we cover a range of materials with mechanical rigidity that can be reversibly tuned using one of several stimuli (e.g. heat, electrical current, electric field, magnetism, etc.). We explain the mechanisms by which these materials change rigidity and how they have been used for robot tasks. We quantitatively assess the performance in terms of the magnitude of rigidity, variation ratio, response time, and energy consumption, and explore the correlations between these desired characteristics as principles for material design and usage.