The Jacobian Conjecture is one of the most famous open problems in mathematics. It asserts the following: if a polynomial map from complex n-space to itself is such that the determinant of the Jacobian matrix is a no-zero constant, then the map is invertible. It is a generalization of the corresponding well-known result from linear algebra. The problem is open for all n>1, inspite of the effort of many mathematicians in almost 80 years! Recently, in 2009 the Chinese mathematician Wenhua Zhao introduced a new concept in algebra, the so-called Mathieu subspace of a ring, which generalizes the notion of an ideal in a ring which was introduced by Emmy Noether in 1922.

In this talk we will introduce this new concept and illustrate it with several examples. Then we will show that the Jacobian Conjecture can be studied in the framework of these Mathieu-Zhao spaces and describe a chain of challenging conjectures, all implying the Jacobian Conjecture. This talk is aimed at a general mathematical audience.