  We describe new optically thin solutions for rotating accretion flows around
black holes and neutron stars. These solutions are advection dominated, so that
most of the viscously dissipated energy is advected radially with the flow. We
model the accreting gas as a two temperature plasma and include cooling by
bremsstrahlung, synchrotron, and Comptonization. We obtain electron
temperatures $T_e\sim 10^{8.5}-10^{10}$K. The new solutions are present only
for mass accretion rates $\dot M$ less than a critical rate $\dot M_{crit}$
which we calculate as a function of radius $R$ and viscosity parameter
$\alpha$. For $\dot M<\dot M_{crit}$ we show that there are three equilibrium
branches of solutions. One of the branches corresponds to a cool optically
thick flow which is the well-known thin disk solution of Shakura \& Sunyaev
(1973). Another branch corresponds to a hot optically thin flow, discovered
originally by Shapiro, Lightman \& Eardley (1976, SLE). This solution is
thermally unstable. The third branch corresponds to our new advection-dominated
solution. This solution is hotter and more optically thin than the SLE
solution, but is viscously and thermally stable. It is related to the ion torus
model of Rees et al. (1982) and may potentially explain the hard X-ray and
$\gamma$-ray emission from X-ray binaries and active galactic nuclei.
