The maximum stresses to which a wide range of mammalian limb tendons could be subjected in life were estimated by considering the relative cross-sectional areas of each tendon and of the fibres of its muscle. These cross-sectional areas were derived from mass and length measurements on tendons and muscles assuming published values for the respective densities. The majority of the stresses are low. The distribution has a broad peak with maximum frequency at a stress of about 13 MPa, whereas the fracture stress for tendon in tension is about 100 MPa. Thus, the majority of tendons are far thicker than is necessary for adequate strength. Much higher stresses are found among those tendons which act as springs to store energy during locomotion. The acceptability of low safety factors in these tendons has been explained previously (Alexander, 1981). A new theory explains the thickness of the majority of tendons. The muscle with its tendon is considered as a combined system which delivers mechanical energy: the thickness of the tendon is optimized by minimizing the combined mass. A thinner tendon would stretch more. To take up this stretch, the muscle would require longer muscle fibres, which would increase the combined mass. The predicted maximum stress in a tendon of optimum thickness is about 10 MPa, which is within the main peak of the observed stress distribution. Individual variations from this value are to be expected and can be understood in terms of the functions of the various muscles.