  The purpose of the present paper is twofold: to introduce the notion of a
generalized flag in an infinite dimensional vector space $V$ (extending the
notion of a flag of subspaces in a vector space), and to give a geometric
realization of homogeneous spaces of the ind--groups $SL(\infty)$, $SO(\infty)$
and $Sp(\infty)$ in terms of generalized flags. Generalized flags in $V$ are
chains of subspaces which in general cannot be enumerated by integers. Given a
basis $E$ of $V$, we define a notion of $E$--commensurability for generalized
flags, and prove that the set $\cFl (\cF, E)$ of generalized flags
E$--commensurable with a fixed generalized flag $\cF$ in $V$ has a natural
structure of an ind--variety. In the case when $V$ is the standard
representation of $G = SL(\infty)$, all homogeneous ind--spaces $G/P$ for
parabolic subgroups $P$ containing a fixed splitting Cartan subgroup of $G$,
are of the form $\cFl (\cF, E)$. We also consider isotropic generalized flags.
The corresponding ind--spaces are homogeneous spaces for $SO(\infty)$ and
$Sp(\infty)$. As an application of the construction, we compute the Picard
group of $\cFl (\cF, E)$ (and of its isotropic analogs) and show that $\cFl
(\cF, E)$ is a projective ind--variety if and only if $\cF$ is a usual,
possibly infinite, flag of subspaces in $V$.
