YIELDING of materials is not understood well enough for detailed, quantitative predictions of strength to be possible, except by using semi-empirical models 1,2. Studies of material strength at high pressures are therefore of fundamental as well as practical interest for determining the relationship between strength and other physical properties3-6. To this end, we have measured the shear stress (tau) supported by rhenium at pressures of up to 120 GPa, far higher than the pressures used in previous studies. Rhenium is of particular interest because it has the highest known bulk and shear moduli among metallic elements 7-9. By using two independent methods determining shear stress at room temperature, we find that rhenium is one of the strongest polycrystalline materials investigated so far, with shear stresses at high pressures reaching tau/mu almost-equal-to 0.04 (+/- 0.02) relative to the shear modulus-mu. These values of tau/mu are nevertheless compatible with current theoretical expectations, indicating that the high strength of rhenium is not anomalous 1,2,6.