  We give an introduction into some aspects of the emerging mathematical theory
of many-body localization (MBL) for disordered quantum spin chains. In
particular, we discuss manifestations of MBL such as zero-velocity
Lieb-Robinson bounds, quasi-locality of the time evolution of local
observables, as well as exponential clustering and low entanglement of
eigenstates. Explicit models where such properties have recently been verified
are the XY and XXZ spin chain, in each case with disorder introduced in the
form of a random exterior field. We introduce these models, state many of the
available results and try to provide some general context. We discuss methods
and ideas which enter the proofs and, in a few illustrative examples, include
more detailed arguments. Finally, we also mention some directions for future
mathematical work on MBL.
