In a low-mass X-ray binary (LMXB), an intense stellar wind from the mass donor may be a consequence of the absorption of X-rays from the mass-accreting neutron star or black hole, and such a wind could change the evolution of these binaries dramatically compared with the evolution of cataclysmic variables (CVs), which are close binaries in which the accretor is a white dwarf. An analytical study and numerical models show that, in the closest and brightest LMXBs, a relativistic companion can capture up to ~10% of the mass lost in the induced stellar wind (ISW) from the main-sequence or subgiant donor, and this is enough to keep the X-ray luminosity of a typical LMXB on the level of LX ~ 5000 L☉ and to accelerate the rotation of an old neutron star with a low magnetic field into the millisecond-period range. A self-sustained ISW may exist even if the donor does not fill its Roche lobe, but the system can be bright (LX > 100 L☉) only if the radius of the donor is a substantial fraction ( 0.8) of the Roche lobe radius. A lower limit on the Roche lobe filling factor follows from the circumstance that both the rate Ėwind at which work must be done to lift wind matter off the donor and the rate Ėabs at which the donor absorbs X-ray energy are proportional to ISW (the ISW mass-loss rate) and from the requirement that Ėwind 3 hr. In Algol-like LMXBs in the Galactic disk, the timescale for the evaporation (caused by the ISW) of the donor with a low-mass, degenerate helium core can be smaller than the timescale for the radial expansion of the donor owing to nuclear evolution, and the donor may never fill its Roche lobe. However, if progenitor binaries are initially wide enough, the donor may escape evaporation as a main-sequence star, and significant mass transfer may not occur until the secondary evolves into a giant with a degenerate helium core of large mass and fills its Roche lobe. In globular clusters, as a result of capture and exchange reactions, semidetached Algol-like LMXBs can be formed in which the donor can fill its Roche lobe even when its degenerate helium core is of small mass, and Roche lobe mediated mass transfer driven by the nuclear evolution of the donor can dominate over capture from the ISW. The numerical models formally imply the possible presence in the Galaxy of ~104 dim (LX ~ 1-100 L☉), long-period LMXBs or radio pulsars with low-mass ( ~0.05 M☉) companions. Since there are few, if any, known observational counterparts of these systems, it is necessary to invoke a mechanism or mechanisms to destroy their formal progenitors. Possible destruction mechanisms include: (1) evaporation driven by the radiation from the rapidly rotating pulsar into which the accretor has been transformed by accretion during the bright LMXB phase, and (2) a dynamical instability arising when the donor is almost completely convective and fills its Roche lobe. In the case of dynamical disruption, the donor may be transformed into the envelope of a Thorne-Żytkow (1975) object with a neutron star or black hole core or into a planet-forming disk around the neutron star or black hole. A few short-period (Porb < 3 hr) LMXBs do exist, and, in them, the donor may be a helium white dwarf of mass less than ~0.09 M☉. An ISW operating before the donor fills its Roche lobe may be responsible for reducing the mass of the white dwarf from an initial value of ≥0.13 M☉ to a value of ≤0.09 M☉, thus permitting stable mass exchange (at a rate smaller than the Eddington limiting rate) and evolution to longer periods to occur after the donor fills its Roche lobe. Another scenario relies on the collapse of a massive oxygen-neon white dwarf, which has accreted from a Roche lobe filling helium white dwarf. Problems that must be explored further in order to acquire a better understanding of the evolution of LMXBs include the formation of a corona around an irradiated low-mass main-sequence or degenerate dwarf star, accretion of ISW matter by a neutron star or black hole companion, the effect of an ISW on the MSW, formation of millisecond pulsars, complete evaporation of low-mass donors, disruption by tidal forces of a low-mass main-sequence star or a degenerate dwarf companion into a gas disk around the accretor, and the formation of planetary systems in the disk around neutron stars and or black holes in post-LMXB systems.