We will discuss recent progress in the construction and classificationof infinite dimensional modules M over a complex semisimple Lie algebra,g. We say that M is a generalized Harish-Chandra module if for some subalgebra k reductive in g, M is a sum of finite dimensional
k-submodules, each occuring with finite multiplicity.
The class of generalized Harish-Chandra modules includes Verma modules,weight modules with finite weight multiplicities, and Harish-Chandra modules, for which k can be chosen to be the fixed points of an involution of g. Not all simple g-modules are generalized-Harish Chandra modules, and not all generalized Harish-Chandra modules are
isomorphic to either weight modules with finite weight multiplicities or Harish-Chandra modules. Thus, the study of generalized Harish-Chandra modules constitutes a new and potentially tractible domain in representation theory.