Quantum spin systems are many-body models which are of wide interest in modern physics and at the same time amenable to rigorous mathematical analysis. A central question about a quantum spin system is whether its Hamiltonian exhibits a spectral gap above the ground state. The existence of such a spectral gap has far-reaching consequences, e.g., for the ground state complexity. In this talk, we survey recent progress on deriving spectral gaps for frustration-free quantum spin systems in dimensions greater than 1, in particular, in the antiferromagnetic models of Affleck-Kennedy-Lieb-Tasaki (AKLT).