  In this paper, we study some qualitative properties of
Hardy-Littlewood-Sobolev type systems. The HLS type systems are categorized
into three cases: critical, supercritical and subcritical. The critical case,
the well known original HLS system, corresponds to the Euler-Lagrange equations
of the fundamental HLS inequality. In each case, we give a brief survey on some
important results and useful methods. Some simplifications and extensions based
on somewhat more direct and intuitive ideas are presented. Also, a few new
qualitative properties are obtained and several open problems are raised for
future research.
