Most attempts to generalize the Jacobian Conjecture failed. The only such an attempt which is still standing is a conjecture due to Olivier Mathieu from 1995, concerning integrals over compact connected Lie groups. However this conjecture is extremely hard. The proof of the Abelian case, due to Duistermaat and van der Kallen, is already a tour de force. In 2009, Wenhua Zhao came up with a new set of fascinating conjectures. One of them, the Image Conjecture, implies the Jacobian Conjecture. Both Mathieus conjecture and the Image Conjecture are embedded in the context of Mathieu subspaces, a concept introduced by Zhao which generalizes the notion of an ideal. Now for the first time in the long history of the Jacobian Conjecture there is a beautiful general context in which this conjecture can be studied. Relations with orthogonal polynomials and differential operators give new oppertunities. In this proposal we intend to study examples of Mathieu subspaces, including the ones directly related to the Jacobian conjecture and, starting from these examples, try to lay foundations for the development of a theory of Mathieu subspaces.