  We survey results arising from the study of domains in C^n with non-compact
automorphism group. Beginning with a well-known characterization of the unit
ball, we develop ideas toward a consideration of weakly pseudoconvex (and even
non-pseudoconvex) domains with particular emphasis on characterizations of (i)
smoothly bounded domains with non-compact automorphism group and (ii) the Levi
geometry of boundary orbit accumulation points.
  Particular attention will be paid to results derived in the past ten years.
