A density-functional theory of the isotropic-nematic phase transition in both rigid and semiflexible hard-sphere chain fluids is described. The theory is based on an exact analytical evaluation of the excluded volume and second virial coefficient B-2 for rigid chain molecules, which demonstrates that B-2 in these cases is equivalent to that of a binary mixture of hard spheres and hard diatomic molecules. It is assumed that the same binary-mixture representation applies to semiflexible chains, while scaled particle theory is used to obtain the properties of the fluid at arbitrary densities. The results of the theory are in very good agreement with Monte Carlo (MC) simulation data for rigid tangent hard-sphere chains, but in lesser agreement with available MC studies of rigid fused hard-sphere chains. We find that there is reasonable agreement between the theory and MC data for semiflexible tangent chains, which improves with increasing chain length. The behavior predicted by the theory for semiflexible chains is contrasted with that given by the Khokhlov and Semenov theory of nematic ordering of wormlike polymer chains.