We carry out a detailed study of exclusive radiative rareB-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. Our calculation is restricted to the leading twist-two operators. The decays considered are:B u,d rarrK* +gamma,B u,d rarrp+gamma,B d rarrohgr+gamma and the corresponding decays of theB s mesons,B s rarrphiv+gamma andB s rarrK*+gamma. Based on our estimate of the transition form factorF 1 BrarrK* (0)=0.32±0.05, we find for the branching ratio BR(BrarrK *+gamma)=(4.8±1.5)×10–5, which is in agreement with the observed value of (4.5±1.5±0.9)×10–5 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 radiativeB-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 radiative transition form factors on theb-quark mass and on the momentum transfer, as well as their interrelation with the CKM-suppressed semileptonic decay form factors in Brarrrgr+ell+vell, which we also calculate in our approach.