symplectomorphisms; etc. These pseudogroups should be thought as the set of the local symmetries of these structures. Pseudogroups of symmetries and geometric structures Manifolds with additional structures can often be defined using the pseudogroups of symmetries of a fixed local model. More precisely, given a pseudogroup , a -atlas on a topological space consists of a standard atlas on such that the changes of coordinates (i.e. the transition maps) belong to . An equivalent class of Γ-atlases is also called a -structure on . In particular, when is the pseudogroup of all locally defined diffeomorphisms of Rn, one recovers the standard notion of a smooth atlas and a smooth structure. More generally, one can define the following objects as -structures on a topological space : flat Riemannian structures, for pseudogroups of isometries of Rn with the canonical Euclidean metric; symplectic structures, for the pseudogroup of symplectomorphisms of R2n with the canonical symplectic form; analytic structures, for the pseudogroup of (real-)analytic diffeomorphisms of Rn; Riemann surfaces, for the pseudogroup of invertible holomorphic functions of a complex variable. More generally, any integrable -structure and any (, )-manifold are special cases of -structures, for suitable pseudogroups . Pseudogroups and Lie theory In general, pseudogroups were studied as a possible theory of infinite-dimensional Lie groups. The concept of a local Lie group, namely a pseudogroup of functions defined in neighbourhoods of the origin of a Euclidean space , is actually closer to Lie's original concept of Lie group, in the case where the transformations involved depend on a finite number of parameters, than the contemporary definition via manifolds. One of Cartan's achievements was to clarify the points involved, including the point that a local Lie group always gives rise to a global group, in the current sense (an analogue of Lie's third theorem, on Lie algebras determining a group). The formal group is yet another approach to the specification of Lie groups, infinitesimally. It is known, however, that local topological groups do not necessarily have global counterparts. Examples of infinite-dimensional pseudogroups abound, beginning with the pseudogroup of all diffeomorphisms of . The interest is mainly in sub-pseudogroups of the diffeomorphisms, and therefore with objects that have a Lie algebra analogue of vector fields. Methods proposed by Lie and by Cartan for studying these objects have become more practical given the progress of computer algebra. In the 1950s, Cartan's theory was reformulated by Shiing-Shen Chern,