The paper discusses new optically thin solutions for rotating accretion flows around black holes and neutron stars, focusing on advection-dominated accretion. These solutions are characterized by most of the viscously dissipated energy being advected radially with the flow. The authors model the accreting gas as a two-temperature plasma and include cooling processes such as bremsstrahlung, synchrotron, and Comptonization. They find that for mass accretion rates below a critical value, there are three equilibrium branches of solutions: a cool optically thick flow (the Shakura-Sunyaev thin disk solution), a hot optically thin flow (the SLE solution), and a new advection-dominated solution. The advection-dominated solution is hotter and more optically thin than the SLE solution but is both thermally and viscously stable. This solution is related to the ion torus model and may explain hard X-ray and $\gamma$-ray emission from X-ray binaries and active galactic nuclei. The paper also highlights differences between advection-dominated accretion around black holes and neutron stars. For black holes, the advected energy is lost into the hole, while for neutron stars, it is thermalized and reradiated at the stellar surface, providing soft photons for Compton cooling. The critical mass accretion rate for black holes is found to be $\dot{M}_{crit} \sim \alpha^2 \dot{M}_{Edd}$, independent of the black hole mass, whereas for neutron stars, it is $\sim 0.1 \alpha^2 \dot{M}_{Edd}$. Advection-dominated accretion is more likely to occur in black holes, leading to underluminous systems due to the bulk of the energy being advected into the hole. The authors argue that in certain circumstances, the advection-dominated solution is the only truly stable state, and a thin disk will spontaneously evaporate and convert into an advection-dominated flow. Even for $\dot{M} > \dot{M}_{crit}$, a thin disk may partially evaporate, with a fraction of the accretion occurring via an advection-dominated hot corona. The paper concludes by discussing the implications of these findings for the properties of accretion flows in black hole and neutron star systems.The paper discusses new optically thin solutions for rotating accretion flows around black holes and neutron stars, focusing on advection-dominated accretion. These solutions are characterized by most of the viscously dissipated energy being advected radially with the flow. The authors model the accreting gas as a two-temperature plasma and include cooling processes such as bremsstrahlung, synchrotron, and Comptonization. They find that for mass accretion rates below a critical value, there are three equilibrium branches of solutions: a cool optically thick flow (the Shakura-Sunyaev thin disk solution), a hot optically thin flow (the SLE solution), and a new advection-dominated solution. The advection-dominated solution is hotter and more optically thin than the SLE solution but is both thermally and viscously stable. This solution is related to the ion torus model and may explain hard X-ray and $\gamma$-ray emission from X-ray binaries and active galactic nuclei. The paper also highlights differences between advection-dominated accretion around black holes and neutron stars. For black holes, the advected energy is lost into the hole, while for neutron stars, it is thermalized and reradiated at the stellar surface, providing soft photons for Compton cooling. The critical mass accretion rate for black holes is found to be $\dot{M}_{crit} \sim \alpha^2 \dot{M}_{Edd}$, independent of the black hole mass, whereas for neutron stars, it is $\sim 0.1 \alpha^2 \dot{M}_{Edd}$. Advection-dominated accretion is more likely to occur in black holes, leading to underluminous systems due to the bulk of the energy being advected into the hole. The authors argue that in certain circumstances, the advection-dominated solution is the only truly stable state, and a thin disk will spontaneously evaporate and convert into an advection-dominated flow. Even for $\dot{M} > \dot{M}_{crit}$, a thin disk may partially evaporate, with a fraction of the accretion occurring via an advection-dominated hot corona. The paper concludes by discussing the implications of these findings for the properties of accretion flows in black hole and neutron star systems.