The aim of this paper is to show that the theory of jet bundles supplies the appropriate setting for the study of Backlund trans- formations. These transformations are used to solve certain partial differential equations, particularly non-linear evolution equations. Of course jets have been employed for some time in the theory of partial differential equations, but so far little use has been made of them in applications. In the meanwhile, substantial progress has been made in the study of non-linear evolution equations. This work has been encouraged by the dis- covery of remarkable properties...