With EPR and ENDOR hyperfine interactions with silicon and impurity nuclei have been determined for many different defects and impurity centers in silicon. A standard interpretation is mostly given with an LCAO description, as introduced into this field by Watkins and Corbett. Usually, this method only includes contributions from atomic orbitals centered on the pertinent nuclei. The anisotropic part of the hyperfine interaction arises from dipole-dipole interaction due to p-(or d-)orbitals. Only under special circumstances perceptible contributions are made by dipole-dipole interaction between nuclei and unpaired spin density which is localized at other nuclei. If this is the case, additional information on the detailed structure of the defects can be obtained. Several examples are discussed in this paper, both for vacancy-related defects and for transition metal impurities in silicon. For the vacancy we derive a lattice relaxation which is opposite to theoretical predictions.