We propose an extension of the concept of Fucik spectrum to the case of coupled systems of two elliptic equations, we study its structure and some applications. We show that near a simple eigenvalue of the system, the Fucik spectrum consists (after a suitable reparametrization) of two (maybe coincident) 2-dimensional surfaces. Furthermore, by variational methods, parts of the Fucik spectrum which lie far away from the diagonal (i.e. from the eigenvalues) are found. As application, some existence, non-existence and multiplicity results to systems with eigenvalue crossing (``jumping'') nonlinearities are proved.