Possible scenarios are discussed for the evolution of close binaries characterized primordially by: masses of individual components less than ∼10 M⊙, semimajoraxes A in the range 10≲A/R⊙≲2000, and orbital periodsPorbless than 25 yrs. Final products include double degenerate dwarfs withA≳(3‐5)R⊙, single rapidly rotating dwarfs (Prot≳10 sec), or neutron stars (Prot≳10−3sec), and supernovae of type I (SNeI).We suggest that binary systems consisting of a low mass non‐degenerate component and a neturon star or black hole may be primarily the result of inelastic capture collisions between single neutron stars and single low mass stars or of exchange captures involving the replacement of one component of a low mass binary by a passing neutron star. Tidal capture and exchange capture processes can occur with the necessary frequency (∼10−7yr−1) in high stellar density regions in globular clusters, in the Galactic bulge, and possibly also in the Galactic disk. The evolution of such systems leads ultimately to the formation of massive (≳1.4 M⊙) and rapidly rotating (Prot≳10−3sec) single neutron stars or to binaries containing a neutron star and a degenerate dwarf.A simple conservation law for ‘‘accumulation’’ instabilities accompanying the evolution of low mass binaries is presented. Several accumulation instabilities leading to SNeI and novae, to dwarf novae and X‐ray recurrent bursters, and to X‐ray and &ggr;‐ray bursters are discussed. It is shown that, from considerations both of evolution and of energetics, &ggr;‐ray bursters cannot be explained by systems consisting of a neturon star with a low mass companion. They can, however, be understood in terms of either of two models in which a single young (age ≳107yr) neutron star with a strong magnetic field is accreting matter from (1) the interstellar medium or (2) from the wind emitted by an OB star companion which does not fill its Roche lobe.