  Let $\frak g$ be a reductive Lie algebra over $\bold C$. We say that a $\frak
g$-module $M$ is a generalized Harish-Chandra module if, for some subalgebra
$\frak k \subset\frak g$, $M$ is locally $\frak k$-finite and has finite $\frak
k$-multiplicities. We believe that the problem of classifying all irreducible
generalized Harish-Chandra modules could be tractable. In this paper, we review
the recent success with the case when $\frak k$ is a Cartan subalgebra. We also
review the recent determination of which reductive in $\frak g$ subalgebras
$\frak k$ are essential to a classification. Finally, we present in detail the
emerging picture for the case when $\frak k$ is a principal 3-dimensional
subalgebra.
