  In this contribution, a mathematical framework is constructed to relate and
compare non-linear partial differential equations (PDEs) in the category of
smooth manifolds. In particular, it can be used to compare those aspects of
field theories (e.g. of classical (Newtonian) mechanics, hydrodynamics,
electrodynamics, relativity theory, classical Yang-Mills theory and so on) that
are described by such equations. Employing a geometric (jet space) approach, a
suitable notion of shared structure of two systems of PDEs is identified. It is
proven that this shared structure can serve to transfer solutions from one
theory to another and a generalization of so-called B\"acklund transformations
is derived that can be used to generate non-trivial solutions of some
non-linear PDEs. A procedure (based on formal integrability) is introduced with
which one can explicitly compute the minimal consistency conditions that two
systems of PDEs need to fulfill in order to share structure under a given
correspondence. Furthermore, it is shown how symmetry groups can be used to
identify useful correspondences and structure that is shared up to symmetries.
Thereby, the role that B\"acklund transformations play in the theory of
quotient equations is clarified. Explicit examples illustrate the general ideas
throughout the text and in the last chapter, the framework is applied to
systems related to electrodynamics and hydrodynamics.
