  This is a review of results on the structure of the homogeneous ind-varieties
$G/P$ of the ind-groups $G=\mathrm{GL}_{\infty}(\mathbb{C})$,
$\mathrm{SL}_{\infty}(\mathbb{C})$, $\mathrm{SO}_{\infty}(\mathbb{C})$,
$\mathrm{Sp}_{\infty}(\mathbb{C})$, subject to the condition that $G/P$ is a
inductive limit of compact homogeneous spaces $G_n/P_n$. In this case the
subgroup $P\subset G$ is a splitting parabolic subgroup of $G$, and the
ind-variety $G/P$ admits a "flag realization". Instead of ordinary flags, one
considers generalized flags which are, generally infinite, chains $\mathcal{C}$
of subspaces in the natural representation $V$ of $G$ which satisfy a certain
condition: roughly speaking, for each nonzero vector $v$ of $V$ there must be a
largest space in $\mathcal{C}$ which does not contain $v$, and a smallest space
in $\mathcal{C}$ which contains $v$.
  We start with a review of the construction of the ind-varieties of
generalized flags, and then show that these ind-varieties are homogeneous
ind-spaces of the form $G/P$ for splitting parabolic ind-subgroups $P\subset
G$. We also briefly review the characterization of more general, i.e.
non-splitting, parabolic ind-subgroups in terms of generalized flags. In the
special case of an ind-grassmannian $X$, we give a purely algebraic-geometric
construction of $X$. Further topics discussed are the Bott--Borel--Weil Theorem
for ind-varieties of generalized flags, finite-rank vector bundles on
ind-varieties of generalized flags, the theory of Schubert decomposition of
$G/P$ for arbitrary splitting parabolic ind-subgroups $P\subset G$, as well as
the orbits of real forms on $G/P$ for $G=\mathrm{SL}_{\infty}(\mathbb{C})$.
