Advection-dominated accretion flows around black holes and neutron stars are described in this paper. These flows are characterized by the advection of viscously dissipated energy radially with the flow, rather than radiative cooling. The accreting gas is modeled as a two-temperature plasma, with cooling via bremsstrahlung, synchrotron, and Comptonization. Electron temperatures range from $10^{8.5}$ to $10^{10}$ K. The paper presents new solutions for advection-dominated accretion flows, which exist only for mass accretion rates below a critical value $ \dot{M}_{crit} $, which depends on radius and viscosity parameter $ \alpha $. For $ \dot{M} < \dot{M}_{crit} $, three equilibrium branches of solutions are found: a cool optically thick flow (thin disk solution), a hot optically thin flow (SLE solution), and a new advection-dominated solution. The latter is hotter and more optically thin than the SLE solution, but is both thermally and viscously stable. It is related to the ion torus model and may explain hard X-ray and gamma-ray emission from X-ray binaries and AGN. For $ \dot{M} < \dot{M}_{crit} $, the accretion flow can choose between the thin disk solution and the new advection-dominated solution. The latter is more stable and may be the preferred state in certain circumstances. Even for $ \dot{M} > \dot{M}_{crit} $, a thin disk may partially evaporate, with some accretion occurring via an advection-dominated hot corona. These ideas suggest that optically thin advection-dominated flows are widespread, possibly the most common form of accretion in black holes at sub-Eddington rates. Advection-dominated accretion around black holes differs from similar flows around neutron stars. In the former, advected energy is lost into the black hole, while in the latter, it is thermalized and reradiated at the stellar surface, providing soft photons that can Compton-cool the gas. The critical accretion rate $ \dot{M}_{crit} $ is $ \sim \alpha^{2} \dot{M}_{Edd} $ for black holes and $ \sim 0.1\alpha^{2} \dot{M}_{Edd} $ for neutron stars. Advection-dominated accretion is therefore more likely in black holes, and these systems will be underluminous for their $ \dot{M} $ because the bulk of the energy is advected into the hole rather than being radiated. Electron temperatures in black hole flows rise up to $ \sim 10^{9} - 10^{10} $ K, compared to $ \sim 10^{8.5}Advection-dominated accretion flows around black holes and neutron stars are described in this paper. These flows are characterized by the advection of viscously dissipated energy radially with the flow, rather than radiative cooling. The accreting gas is modeled as a two-temperature plasma, with cooling via bremsstrahlung, synchrotron, and Comptonization. Electron temperatures range from $10^{8.5}$ to $10^{10}$ K. The paper presents new solutions for advection-dominated accretion flows, which exist only for mass accretion rates below a critical value $ \dot{M}_{crit} $, which depends on radius and viscosity parameter $ \alpha $. For $ \dot{M} < \dot{M}_{crit} $, three equilibrium branches of solutions are found: a cool optically thick flow (thin disk solution), a hot optically thin flow (SLE solution), and a new advection-dominated solution. The latter is hotter and more optically thin than the SLE solution, but is both thermally and viscously stable. It is related to the ion torus model and may explain hard X-ray and gamma-ray emission from X-ray binaries and AGN. For $ \dot{M} < \dot{M}_{crit} $, the accretion flow can choose between the thin disk solution and the new advection-dominated solution. The latter is more stable and may be the preferred state in certain circumstances. Even for $ \dot{M} > \dot{M}_{crit} $, a thin disk may partially evaporate, with some accretion occurring via an advection-dominated hot corona. These ideas suggest that optically thin advection-dominated flows are widespread, possibly the most common form of accretion in black holes at sub-Eddington rates. Advection-dominated accretion around black holes differs from similar flows around neutron stars. In the former, advected energy is lost into the black hole, while in the latter, it is thermalized and reradiated at the stellar surface, providing soft photons that can Compton-cool the gas. The critical accretion rate $ \dot{M}_{crit} $ is $ \sim \alpha^{2} \dot{M}_{Edd} $ for black holes and $ \sim 0.1\alpha^{2} \dot{M}_{Edd} $ for neutron stars. Advection-dominated accretion is therefore more likely in black holes, and these systems will be underluminous for their $ \dot{M} $ because the bulk of the energy is advected into the hole rather than being radiated. Electron temperatures in black hole flows rise up to $ \sim 10^{9} - 10^{10} $ K, compared to $ \sim 10^{8.5}