  We revisit the entire framework of group classification of differential
equations. After introducing the notion of weakly similar classes of
differential equations, we develop the mapping method of group classification
for such classes, which generalizes all the versions of this method that have
been presented in the literature. The mapping method is applied to group
classification of various classes of Kolmogorov equations and of Fokker-Planck
equations in the case of space dimension one. The equivalence groupoids and the
equivalence groups of these classes are computed. The group classification
problems for these classes with respect to the corresponding equivalence groups
are reduced to finding all inequivalent solutions of heat equations with
inequivalent potentials admitting Lie-symmetry extensions. This reduction
allows us to exhaustively solve the group classification problems for the
classes of Kolmogorov and Fokker-Planck equations with time-independent
coefficients.
