## 1 Introduction  
The metal-to-insulator transition (MIT) is one of the most spectacular effects in condensed matter physics and materials science. The dramatic change in electrical properties of materials undergoing such a transition is exploited in electronic devices that are components of data storage and memory technology [1; 2]. It is generally recognized that the underlying mechanism of MITs are the interplay of electron correlation effects (Mott type) and disorder effects (Anderson type) [3; 4; 5; 6; 7]. Recent developments in many-body physics make it possible to study these phenomena on equal footing rather than having to disentangle the two.  
The purpose of this review is to bring together the various developments and applications of such a new method, namely the Typical Medium Dynamical Cluster Approach (TMDCA) [8; 9; 10; 11; 12], for investigating interacting disordered quantum systems.  
The organization of this article is as follows: Section 2 is dedicated to a few basic aspects of modeling disorder in solids. We discuss a couple of examples of materials that are believed to have relevant technological applications connected to the problem of localization. The corresponding subsections deal with theoretical modeling. We then follow with a review of the Anderson and Mott mechanisms leading to electronic localization, as well as their interplay.  
In Section 3 we review three alternative numerical methods for solving the Anderson model and discuss their advantages and limitations in chemically specific modeling. These methods are employed in Section 7 to validate the developed formalism.  
In Section 4 we shift our focus to the discussion of the effective medium methods. First, we present the concept of coarse-graining. The coarse-graining procedure allows us to draw similarities present in infinite dimension between the Dynamical Mean-Field Theory (DMFT) [13; 14; 15; 16; 17; 18; 19] of interacting electrons and the Coherent Potential Approximation (CPA) [20; 21; 22] of non-interacting electrons in disordered external potentials. We then provide a detailed discussion of the Dynamical Cluster Approximation [8; 23; 24], a non-local effective medium approximation, which systematically incorporates the non-local correlation effects missing in the DMFT and CPA by refining the course graining.  
The central focus of this review is the typical medium theories of Anderson localization, which are discussed in Section 5. We show how this method is used to study disorder-induced electron localization. Starting from the single-site typical medium theory, we present its natural cluster extension, discussing several algorithms for the self-consistent embedding of periodic clusters fulfilling the original symmetries of the lattice in addition to other desirable properties. We present details of how this method can be used to incorporate the full chemical complexity of various systems, including off-diagonal disorder and multiband nature, along with the interplay of disorder and electron-electron interactions.  
In Section 6 we discuss how the developed typical medium methods can be practically applied to real materials. This is done in a three-step process in which DFT results are used to generate an effective disordered Hamiltonian, which is passed to the typical medium cluster/single-site solver to compute spectral densities and estimate the degree of localization. Section 7 reviews the application of the TMDCA from single-band three-dimensional models to more complex cases such as off-diagonal disorder, multi-orbital cases, and electronic interactions. Finally, the concluding remarks are presented in Section 8.