  Great progress has been made in the last several years towards understanding
the properties of disordered electronic systems. In part, this is made possible
by recent advances in quantum effective medium methods which enable the study
of disorder and electron-electronic interactions on equal footing. They include
dynamical mean field theory and the coherent potential approximation, and their
cluster extension, the dynamical cluster approximation. Despite their
successes, these methods do not enable the first-principles study of the
strongly disordered regime, including the effects of electronic localization.
The main focus of this review is the recently developed typical medium
dynamical cluster approximation for disordered electronic systems. This method
has been constructed to capture disorder-induced localization, and is based on
a mapping of a lattice onto a quantum cluster embedded in an effective typical
medium, which is determined self-consistently. Here we provide an overview of
various recent applications of the typical medium dynamical cluster
approximation to a variety of models and systems, including single and
multi-band Anderson model, and models with local and off-diagonal disorder. We
then present the application of the method to realistic systems in the
framework of the density functional theory.
