In the light of recent observations in which short gamma-ray bursts are interpreted as arising from black-hole(BH), neutron-star(NS) or NS-NS mergings we would like to review our research on the evolution of compact binaries, especially those containing NS's. These were carried out with predictions for LIGO in mind, but are directly applicable to short gamma-ray bursts in the interpretation above. Most important in our review is that we show that the standard scenario for evolving NS-NS binaries always ends up with a low-mass BH (LMBH), NS binary. Bethe and Brown (1998) showed that this fate could be avoided if the two giants in the progenitor binary burned He at the same time, and that in this way the binary could avoid the common envelope evolution of the NS with red giant companion which sends the first born NS into a BH in the standard scenario. The burning of He at the same time requires, for the more massive giants such as the progenitors of the Hulse-Taylor binary NS that the two giants be within 4% of each other in ZAMS mass. Applying this criterion to all binaries results in a factor 5 of LMBH-NS binaries as compared with NS-NS binaries. Our scenario of NS-NS binaries as having been preceded by a double He-star binary is collecting observational support in terms of the nearly equal NS masses within a given close binary.

32 pages, 1 figure, substantial changes from v1