Exact normal forms in local differential geometry

In the classification problems of local differential geometry the functional dimension of the transformation group is smaller than that of the space of objects, and functional moduli appear. Constructing exact normal form is a difficult task but it is doable with certain modifications of tools of the classical singularity theory. I will explain construction of normal forms in the classification problems of Riemannian metrics, conformal structures, (2,3,5) distributions, and CR geometry, and will give an idea how they can be used for classification of all possible symmetry algebras and finding homogeneous models.