Abstract Accretion disk theory was first developed assuming local heat balance where the energy produced by viscous heating was emitted from the sides of the disk. One of the important inventions of this theory was the phenomenological treatment of the turbulent viscosity, known as the “alpha” prescription, where the ( rφ) component of the stress tensor was approximated by αp, where α is a dimensionless constant and p is the pressure. This prescription played a role in the accretion disk theory as important as the mixing-length theory of convection for stellar evolution. Sources of turbulence in accretion disks are discussed, including hydrodynamical turbulence and convection and the role and of ordered and disordered magnetic fields. In parallel with the optically thick geometrically thin accretion disk models, a new branch of the optically thin accretion disk models was discovered, with a larger thickness for the same total luminosity. The choice between these solutions should be made on the basis of a stability analysis. The ideas underlying the necessity for including advection in accretion disk theory are presented and models with advection are reviewed. The low-luminosity, optically thin accretion disk models with advection are discussed, and the limits on advection dominated accretion flows (ADAF) imposed by the presence of a magnetic field are analyzed. The behavior of accretion disks in the presence of ordered and disordered magnetic fields are discussed. The interaction of an accretion disk with a rotating magnetized star is reviewed including the formation of magnetohydrodynamic (MHD) outflows, magnetic braking of the star’s rotation, and the propeller effect. The problems of the MHD origin of jets from magnetized disks are discussed including the ejection of mass, energy, angular momentum, and magnetic flux from the disk in both in the hydromagnetic and Poynting flux regimes. Recent MHD simulation results are discussed.