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There is mounting evidence for the binary nature of the progenitors ofgamma-ray bursts (GRBs). For a long GRB, the induced gravitational collapse(IGC) paradigm proposes as progenitor, or "in-state", a tight binary systemcomposed of a carbon-oxygen core (CO$_{core}$) undergoing a supernova (SN)explosion which triggers hypercritical accretion onto a neutron star (NS)companion. For a short GRB, a NS-NS merger is traditionally adopted as theprogenitor. We divide long and short GRBs into two sub-classes, depending onwhether or not a black hole (BH) is formed in the merger or in thehypercritical accretion process exceeding the NS critical mass. For longbursts, when no BH is formed we have the sub-class of X-ray flashes (XRFs),with isotropic energy $E_{iso}\lesssim10^{52}$ erg and rest-frame spectral peakenergy $E_{p,i}\lesssim200$ keV. When a BH is formed we have the sub-class ofbinary-driven hypernovae (BdHNe), with $E_{iso}\gtrsim10^{52}$ erg and$E_{p,i}\gtrsim200$ keV. In analogy, short bursts are similarly divided intotwo sub-classes. When no BH is formed, short gamma-ray flashes (S-GRFs) occur,with $E_{iso}\lesssim10^{52}$ erg and $E_{p,i}\lesssim2$ MeV. When a BH isformed, the authentic short GRBs (S-GRBs) occur, with $E_{iso}\gtrsim10^{52}$erg and $E_{p,i}\gtrsim2$ MeV. We give examples and observational signatures ofthese four sub-classes and their rate of occurrence. From their respectiverates it is possible that "in-states" of S-GRFs and S-GRBs originate from the"out-states" of XRFs. We indicate two additional progenitor systems: whitedwarf-NS and BH-NS. These systems have hybrid features between long and shortbursts. In the case of S-GRBs and BdHNe evidence is given of the coincidence ofthe onset of the high energy GeV emission with the birth of a Kerr BH.

On the classification of GRBs and their occurrence rates