  Operator ideals in B(H) are well understood and exploited but ideals inside
them have only recently been studied starting with the 1983 seminal work of
Fong and Radjavi and continuing with two recent articles by the authors of this
survey. This article surveys this study embodied in these three articles. A
subideal is a two-sided ideal of J (for specificity also called a J-ideal) for
J an arbitrary ideal of B(H). In this terminology we alternatively call J a
B(H)-ideal.
  This surveys these three articles in which we developed a complete
characterization of all J-ideals generated by sets of cardinality strictly less
than the cardinality of the continuum. So a central theme is the impact of
generating sets for subideals on their algebraic structure. This
characterization includes in particular finitely and countably generated
J-ideals. It was obtained by first generalizing to arbitrary principal J-ideals
the 1983 work of Fong-Radjavi who determined which principal K(H)-ideals are
also B(H)-ideals. A key property in our investigation turned out to be
J-softness of a B(H)-ideal I inside J, that is, IJ = I, a generalization of a
recent notion of K(H)-softness of B(H)-ideals introduced by Kaftal-Weiss and
earlier exploited for Banach spaces by Mityagin and Pietsch. This study of
subideals and the study of elementary operators with coefficient constraints
are closely related. Here we also introduce and study a notion of
subideal-traces where classical traces (unitarily invariant linear functionals)
need not make sense for subideals that are not B(H)-ideals.
