  We review the current (as of Fall 2016) status of the studies on the emergent
integrability in many-body localized models. We start by explaining how the
phenomenology of fully many-body localized systems can be recovered if one
assumes the existence of a complete set of (quasi)local operators which commute
with the Hamiltonian (local integrals of motion, or LIOMs). We describe the
evolution of this idea from the initial conjecture, to the perturbative
constructions, to the mathematical proof given for a disordered spin chain. We
discuss the proposed numerical algorithms for the construction of LIOMs and the
status of the debate on the existence and nature of such operators in systems
with a many-body mobility edge, and in dimensions larger than one.
