  The stability of mass transfer in binaries with convective giant donors
remains an open question in modern astrophysics. There is a significant
discrepancy between what the existing methods predict for a response to mass
loss of the giant itself, as well as for the mass transfer rate during the
Roche lobe overflow. Here we show that the recombination energy in the
superadiabatic layer plays an important and hitherto unaccounted-for role in he
donor's response to mass loss, in particular on its luminosity and effective
temperature. Our improved optically thick nozzle method to calculate the mass
transfer rate via $L_1$ allows us to evolve binary systems for a substantial
Roche lobe overflow. We propose a new, strengthened criterion for the mass
transfer instability, basing it on whether the donor experiences overflow
through its outer Lagrangian point. We find that with the new criterion, if the
donor has a well-developed outer convective envelope, the critical initial mass
ratio for which a binary would evolve stably through the conservative mass
transfer varies from $1.5$ to $2.2$, which is about twice as large as
previously believed. In underdeveloped giants with shallow convective envelopes
this critical ratio may be even larger. When the convective envelope is still
growing, and in particular for most cases of massive donors, the critical mass
ratio gradually decreases to this value, from that of radiative donors.
