At the end of the nineteen century Norwegian mathematician Sophus Lie (1842-1899), together with Swedish mathematician Albert Bäcklund (1845-1922), established a theorem that in modern terms reads as follows: any Lie transformation (resp., field) of some domain of the jet space Jk(n,m) is either the lift of a contact transformation (resp., field), if m=1, or the lift of a point transformation (resp., field), if m>1. The proof of this pivotal result in the geometric theory of PDEs, which I will carry out in this seminar, gives us the opportunity to examine in depth some features of the jet spaces and their natural structures.