  A generalized F-structure is a complex, isotropic subbundle $E$ of
$T_cM\oplus T^*_cM$ ($T_cM=TM\otimes_{\mathds{R}}\mathds{C}$ and the metric is
defined by pairing) such that $E\cap\bar E^{\perp}=0$. If $E$ is also closed by
the Courant bracket, $E$ is a generalized CRF-structure. We show that a
generalized F-structure is equivalent with a skew-symmetric endomorphism $\Phi$
of $TM\oplus T^*M$ that satisfies the condition $\Phi^3+\Phi=0$ and we express
the CRF-condition by means of the Courant-Nijenhuis torsion of $\Phi$. The
structures that we consider are generalizations of the F-structures defined by
Yano and of the CR (Cauchy-Riemann) structures. We construct generalized
CRF-structures from: a classical F-structure, a pair $(\mathcal{V},\sigma)$
where $\mathcal{V}$ is an integrable subbundle of $TM$ and $\sigma$ is a 2-form
on $M$, a generalized, normal, almost contact structure of codimension $h$. We
show that a generalized complex structure on a manifold $\tilde M$ induces
generalized CRF-structures into some submanifolds $M\subseteq\tilde M$.
Finally, we consider compatible, generalized, Riemannian metrics and we define
generalized CRFK-structures that extend the generalized K\"ahler structures and
are equivalent with quadruples $(\gamma,F_+,F_-,\psi)$, where $(\gamma,F_\pm)$
are classical, metric CRF-structures, $\psi$ is a 2-form and some conditions
expressible in terms of the exterior differential $d\psi$ and the
$\gamma$-Levi-Civita covariant derivative $\nabla F_\pm$ hold. If $d\psi=0$,
the conditions reduce to the existence of two partially K\"ahler reductions of
the metric $\gamma$. The paper ends by an Appendix where we define and
characterize generalized Sasakian structures.
