  This work provides an overview of gapped quantum spin systems, including
concepts, techniques, properties, and results. The basic framework and objects
of interest for quantum spin systems are introduced, and the main ideas behind
methods for proving spectral gaps for frustration-free models are outlined.
After reviewing recent progress on several spectral gap conjectures, we discuss
quasi-locality of the Heisenberg dynamics and its utility in proving properties
of gapped quantum spin systems. Lieb-Robinson bounds have played a central role
in establishing exponential decay of ground state correlations, an area law for
one-dimensional systems, a many-body adiabatic theorem, and spectral gap
stability. They also aided in the development of the quasi-adiabatic
continuation, which is a useful for investigating gapped ground state phases,
both of which are also discussed.
