The first principles description of strongly correlated materials (typically materials containing partially filled d- or f-shells) is currently regarded as one of the great challenges in condensed matter physics. Many disjointed approaches to tackle the problem are being pursued in the different subfields of the electronic structure community.
The term 'strongly correlated' typically refers to systems or phenomena in which the single-particle picture and first-order perturbation theory for the self energy fail to describe the electronic properties. This happens typically in materials with open d- or f-shells. The simultaneous presence of localized electrons and itinerant band states, structural, orbital and spin degrees of freedom gives rise to a rich variety of physics and chemistry that makes these materials attractive for a wide range of applications. This richness combined with the considerable challenges posed to our (theoretical) understanding have made strongly correlated systems a hot and fascinating topic.
One of the hallmarks of strongly correlated systems, that is often used in the first principles community to classify a system as such, is the failure of density functional theory (DFT) in the local-density or generalized gradient approximation (LDA or GGA, respectively). This, however, assumes the notion that the observable of interest is the spectral function of the system as e.g. measured by direct or inverse photoemission or another excited state property. If, on the other hand, ground state properties at zero temperature are considered (e.g. lattice constants, bond length, phase stability) LDA and GGA are often remarkably good . In reality, however, the interesting regions in the phase diagram of strongly correlated systems are usually at finite temperature (and possibly finite doping), which immediately illustrates some of the challenges for first principles theories.
A major deficiency of LDA and GGA is the delocalization (or self-interaction) error , which is particularly severe for systems with partially occupied d- or f-states. Hybrid functionals, on the other hand, partly correct the self-interaction error by incorporating a certain portion of exact exchange, which significantly improves the descriptions of d- or f-electron systems [3,4]. The dependence on adjustable parameters, however, remains a concern. A parameter free ground state approach beyond LDA/GGA is to combine exact exchange with correlation in the random-phase approximation (RPA) [5,6]. This has the additional advantage that many-body perturbation theory in the GW approach (see below) gives a spectral function that is consistent with the RPA energy .
Conversely, correlation effects that govern e.g. the physics of localized f-electrons can in principle be treated systematically by dynamical mean field theory (DMFT) [7,8]. The Kondo resonance in Cerium [9, 10] or the paramagnetic (Mott-) insulating phase of Ce2O3  are only two examples of successful DMFT applications. However, several challenges persist in the field, probably the most important being the accurate determination of the many-body (“Hubbard U”) corrections from first principles, and the simultaneous correction of strongly correlated (d- or f-) states by DMFT and itinerant (eg. s- or p-) states by weak-coupling corrections beyond DFT-LDA. A great advantage of DMFT is that it is intrinsically a finite temperature formalism.
Many-body perturbation theory (MBPT) provides an alternative approach for systems with moderate electronic correlations . Hedin's GW approximation to the many-body self energy  is a step towards a systematic ab initio understanding of such systems. The GW approach corresponds to the first order term of a systematic expansion in MBPT  and has become the method of choice for the description of quasiparticle band structures in weakly correlated solids . Through the screened Coulomb interaction W it captures the screening among itinerant electrons while at the same time treating exchange at the exact exchange level. The latter should account for a large part of the interactions among localized d- or f-electrons, with the additional advantage that localized and itinerant states are treated on the same footing. The first promising calculations to open-shell transition metal oxides or lanthanide compounds [15,16] and correlated phenomena like the Kondo resonance in quantum transport [17,18] are emerging, but are -- at least in the strong coupling limit of f-electron compounds -- limited to the magnetic phases. The description of paramagnetic phases and finite temperatures remains an open challenge.Combining DMFT and GW might give the best of both worlds [12,19], but so far only approximate implementations have been demonstrated .
Increasing the level of abstraction leads away from real materials to quantum lattice models. The Hubbard or the Anderson model are prominent examples and are studied intensively to gain a deeper understanding of strongly correlated phenomena. The literature on strongly correlated model systems is too vast to summarize recent progress comprehensively. However, a few examples include kinks in the dispersion of energy bands caused by many-body correlation effects , disorder induced ferromagnetism and the emergence of an alloy Kondo insulator  and a new efficient perturbation theory that incorporates also long-range correlations .