Humans, bushbabies, frogs, locusts, fleas and other animals jump by rapidly extending a pair of legs. Mathematical models are used to investigate the effect muscle properties, leg design and jumping technique have on jump height. Jump height increases with increased isometric force exerted by leg muscles, their maximum shortening speeds and their series compliances. When ground forces are small multiples of body mass (as for humans), countermovement and catapult jumps are about equally high, and both are much better than squat jumps. Vertebrates have not evolved catapult mechanisms and use countermovement jumps instead. When ground forces are large multiples of body mass, catapult jumps (as used by locusts and fleas) are much higher than the other styles of jump could be. Increasing leg mass reduces jump height, but the proximal-to-distal distribution of leg mass has only a minor effect. Longer legs make higher jumps possible and additional leg segments, such as the elongated tarsi of bushbabies and frogs, increase jump height even if overall leg length remains unchanged. The effects of muscle moment arms that change as the leg extends, and of legs designed to work over different ranges of joint angle, are investigated.