The Mott Insulating state is one of the most fundamental consequences of strong electron interaction in concert with quantum mechanics. In its simplest form, it projects the low energy sector onto a magnetic Heisenberg model, which has played a central role over the past sixty years of unveiling the magnetic properties of Mott insulators. However, it has become increasingly clear that new quantum phases of matter can occur in cases where the Heisenberg approximation is not sufficient. This Sinergia proposal aims to unite leading groups in their respective domains of material synthesis, experiments and theory for a focused coherent effort to study new materials beyond the Heisenberg approximation. Specifically, we shall investigate: i) the presence of orbital degeneracy, where an extra local degree of freedom has to be taken into account, ii) the proximity to the metal-insulator transition, where more complicated spin-spin interactions become significant, or iii) systems like iridates, where the effective local spin-1/2 degree of freedom may lead to strongly anisotropic interactions. The initial thrust will be on investigations of iridium-oxides – an emerging material class which has recently attracted considerable attention world-wide.