  We carry out a detailed study of exclusive radiative rare $B$-decays in the
framework of the QCD sum rules on the light cone, which combines the
traditional QCD sum rule technique with the description of final state vector
mesons in terms of the light-cone wave functions of increasing twist. The
decays considered are: $B_{u,d} \to K^* +\gamma, B_{u,d}\to \rho+\gamma, B_d\to
\omega+\gamma$ and the corresponding decays of the $B_s$ mesons, $B_s\to
\phi+\gamma$ and $B_s\to K^*+\gamma$. Based on our estimate of the transition
form factor $F_1^{B \to K^*\pg}(0) =0.32\pm0.05$, we find for the branching
ratio $BR(B \to K^* + \gamma) = (4.8\pm 1.5)\times 10^{-5}$, which is in
agreement with the observed value measured by the CLEO collaboration. We
present detailed estimates for the ratios of the radiative decay form factors,
which are then used to predict the rates for the exclusive radiative B-decays
listed above. This in principle allows the extraction of the CKM matrix element
$|V_{td}|$ from the penguin-dominated CKM-suppressed radiative decays when they
are measured. We give a detailed discussion of the dependence of the form
factors on the $b$-quark mass and on the momentum transfer, as well as their
interrelation with the CKM-suppressed semileptonic decay form factors in $B\to
\rho+\ell+\nu$, which we also calculate in our approach.
