After solving the problem of the structure of Lie groups which Cartan (following Lie) called "finite continuous groups" (or "finite transformation groups"), Cartan posed the similar problem for "infinite continuous groups", which are now called Lie pseudogroups, an infinite-dimensional analogue of Lie groups (there are other infinite generalizations of Lie groups). The Lie pseudogroup considered by Cartan is a set of transformations between subsets of a space that contains the identical transformation and possesses the property that the result of composition of two transformations in this set (whenever this is possible) belongs to the same set. Since the composition of two transformations is not always possible, the set of transformations is not a group (but a groupoid in modern terminology), thus the name pseudogroup. Cartan considered only those transformations of manifolds for which there is no subdivision of manifolds into the classes transposed by the transformations under consideration. Such pseudogroups of transformations are called primitive. Cartan showed that every infinite-dimensional primitive pseudogroup of complex analytic transformations belongs to one of the six classes: 1) the pseudogroup of all analytic transformations of n complex variables; 2) the pseudogroup of all analytic transformations of n complex variables with a constant Jacobian (i.e., transformations that multiply all volumes by the same complex number); 3) the pseudogroup of all analytic transformations of n complex variables whose Jacobian is equal to one (i.e., transformations that preserve volumes); 4) the pseudogroup of all analytic transformations of 2n > 4 complex variables that preserve a certain double integral (the symplectic pseudogroup); 5) the pseudogroup of all analytic transformations of 2n > 4 complex variables that multiply the above-mentioned double integral by a complex function; 6) the pseudogroup of all analytic transformations of 2n + 1 complex variables that multiply a certain form by a complex function (the contact pseudogroup). There are similar classes of pseudogroups for primitive pseudogroups of real transformations defined by analytic functions of real variables.