  Affleck, Kennedy, Lieb, and Tasaki constructed a spin-1 model that is
isotropic in spins and possesses a provable finite gap above the ground state
more than three decades ago. They also constructed models in two dimensions.
Their construction has impacted subsequent research that is still active. In
this review article, we review some selected the progresses, such as magnetic
ordering of the AKLT models, emerging phases under deforming the AKLT
Hamiltonians, symmetry-protected topological order in several AKLT models,
their spectral gap, and applications for quantum computation.
