Variational problems in the theory of differential equations and dynamical systems are problems that allow a geometric formulation; `solutions can be regarded as critical points of an appropriately chosen action function'. A classical topological theory of critical points is Morse Theory. Floer Theory is a more advanced critical point theory for differential equations that is based on Morse Theory and has been successfully applied to complex problems in Symplectic Geometry and Mathematical Physics. For variational problems Floer Theory is an extension of Morse Theory to infinite dimensional problems. This project proposes the application of Floer Theory to partial differential equations and will make the first steps towards an extension of Floer Theory beyond variational problems. A successful extension of Floer Theory to partial differential equations and non-variational problems will have a multitude of applications.