The stability criteria of rapid mass transfer and common-envelope evolution are fundamental in binary star evolution. They determine the mass, mass ratio, and orbital distribution of many important systems, such as X-ray binaries, type Ia supernovae, and merging gravitational-wave sources. In the limit of extremely rapid mass transfer, the response of a donor star in an interacting binary becomes asymptotically one of adiabatic expansion. We built the adiabatic mass-loss model and systematically surveyed the thresholds for dynamical timescale mass transfer over the entire span of possible donor star evolutionary states. Many studies indicate that new mass transfer stability thresholds play an essential role in the formation and properties of double compact object populations and the progenitors of SNe Ia and detectable GW sources. For example, our studies show that the mass transfer in the red giant and the asymptotic giant branch stars and the massive stars can be more stable than previously believed. Consequently, detailed binary population synthesis studies, using updated unstable mass transfer criteria, predict that the non-conservative stable mass transfer may dominate the formation channel of double stellar-mass black holes and can explain the population of the large mass ratio double stellar-mass black holes. Using our updated mass transfer thresholds, binary population thesis studies by Li et al. show that Ge et al.'s results support the observational double white dwarfs quite well.