This article surveys the main results of the author's work, in collaboration with David Vanderbilt and John Bennetto, on the application of the order-N density-matrix approach, together with ab initio methods, to investigate the atomic structure of dislocation cores in the homopolar semiconductors silicon, carbon, and germanium. In these systems, the predominant dislocations are the 30° and the 90° partial dislocations. For the three materials, the nature of the reconstruction at the core of the 90°-partial dislocation is considered. Both the traditional single-period and our recently proposed double-period core structures are investigated. The double-period geometry is found to be the ground-state structure in all three cases. For silicon, we have also investigated in detail the structure and dynamics of point excitations (kinks, solitons, and kink-soliton complexes) in the cores of the 30°partial dislocation and the single-period geometry of the 90° partial. Our calculated formation energies and migration barriers for these excitations are in good agreement with available experimental results. Furthermore, we have examined the reactions by which high-energy kinks relax into low-energy ones by soliton emission.