In Section 3.5.1, the mathematical concept of Euclidean and affine normalizers of space groups is introduced. Some crystallographic problems are mentioned for which the solution of the problem is simplified by the use of normalizers. In Section 3.5.2, the properties of the Euclidean and affine normalizers of the plane groups and the space groups are discussed and described in detailed tables that also take into account the dependence of the Euclidean normalizers on the specialization of the metrical parameters for monoclinic and orthorhombic space groups. In addition, for the first time, chirality-preserving Euclidean normalizers of the space groups are listed. In Section 3.5.3, examples for the use of Euclidean and affine normalizers for crystallographic purposes are given: (i) The derivation of Euclidean- and affine-equivalent point configurations and Wyckoff positions constitutes the basis for the definition of Wyckoff sets. The derivation of all different coordinate descriptions of a certain crystal structure is described provided that the description of its space group (basis vectors and origin) remains unchanged. (ii) Each transition from one coordinate description of a crystal structure to another equivalent one necessarily causes changes in the corresponding list of structure factors: either the phases of the reflections or the phases and the indices are changed. As a consequence, the Euclidean normalizers of the space groups lead to a simple derivation of phase restrictions for use in direct methods to `fix the origin and the enantiomorph'. (iii) Different subgroups (or supergroups) of a given space group that play an analogous role with respect to this space group may be identified with the aid of the Euclidean or affine normalizers. (iv) The ranges of the metrical and coordinate parameters that have to be considered for geometrical studies of point configurations can be reduced with the aid of the Euclidean and affine normalizers of space groups. Finally, in Section 3.5.4, the normalizers of the two-dimensional point groups with respect to the full isometry group of the circle and of the three-dimensional point groups with respect to the full isometry group of the sphere are tabulated.