Transition-metal carbides and nitrides are hard materials widely used for cutting tools and wear-resistant coatings. Their hardness is not yet understood at a fundamental level, A clue may lie in the puzzling fact that transition-metal carbonitrides that have the rock-salt structure (such as TiCxN1-x) have the greatest hardness for a valence-electron concentration of about 8.4 per cell(1-3), which suggests that the hardness may be determined more by the nature of the bonding than by the conventional microstructural features that determine the hardness of structural metals and alloys. To investigate this possibility, we have evaluated the shear modulus of various transition-metal carbides and nitrides using nb initio pseudopotential calculations. Our results show that the behaviour of these materials can be understood on a fundamental level in terms of their electronic band structure. The unusual hardness originates from a particular band of sigma bonding states between the non-metal p orbitals and the metal d orbitals that strongly resists shearing strain or shape change. Filling of these states is completed at a valence-electron concentration of about 8.4, and any additional electrons would go into a higher band which is unstable against shear deformations.