Flexible and shape-shifting structures are common in nature. Creating shape-shifting machines using rigid components requires complex design work. Soft robotic attempts are closer to the muscular hydrostats they mimic but are slow and inefficient untethered and do not generally allow large-scale shape change between forms. Here we present the idea of “tensegristat” robots in which 1) we use tendons drawn by electric motors as active tensile elements, and fluid as a passive compressive element; 2) we rely primarily on buckling for motion and shape change; and (3) the total mass of fluid in the robot remains constant. This structure is analogous to that of muscular hydrostats and allows large-scale shape changes, separate control of stiffness and motion, and fast and efficient untethered operation. We demonstrate this idea with a morphing amphibious robot.