  Accretion disc theory was first developed as a theory with the local heat
balance, where the whole energy produced by a viscous heating was emitted to
the sides of the disc. One of the most important new invention of this theory
was a phenomenological treatment of the turbulent viscosity, known as ''alpha''
prescription, when the (r$\phi$) component of the stress tensor was
approximated by ($\alpha$ P) with a unknown constant $\alpha$. This
prescription played the role in the accretion disc theory as well important as
the mixing-length theory of convection for stellar evolution. Sources of
turbulence in the accretion disc are discussed, including nonlinear
hydrodynamical turbulence, convection and magnetic field role. In parallel to
the optically thick geometrically thin accretion disc models, a new branch of
the optically thin accretion disc models was discovered, with a larger
thickness for the same total luminosity. The choice between these solutions
should be done of the base of a stability analysis. The ideas underlying the
necessity to include advection into the accretion disc theory are presented and
first models with advection are reviewed. The present status of the solution
for a low-luminous optically thin accretion disc model with advection is
discussed and the limits for an advection dominated accretion flows (ADAF)
imposed by the presence of magnetic field are analysed.
