In this talk, I will review the recent results we have obtained on the SU(N) Heisenberg model of quantum permutations, which describes the low-energy properties of a number of Mott insulating phases in condensed matter physics (spin-1 antiferromagnets with biquadratic interaction, symmetric spin-orbital model) and cold atoms (ultracold ferminionic alkaline rare-earths in optical lattices). In 1D, where Quantum Monte Carlo simulations can be performed, we have calculated the correlations as a function of the entropy per site, with the conclusion that the entropy below which characteristic features show up increases with N and reaches experimentally accessible values already for N=4. 
In 2D, using a variety of analytical and numerical approaches, in particular flavour-wave theory and a tensor-network algorithm, we have shown that the nature of the ground state depends crucially on the value of N and on the topology of the lattice, including long-range color order for SU(3) on the triangular and square lattices, spontaneous dimerization for SU(4) on the square lattice, and an algebraic quantum liquid for SU(4) on the honeycomb lattice. Experimental implications for condensed matter systems and cold atoms will be briefly discussed.