2008: Iron-based Superconductors:

The discovery of superconductivity with a Tc of 26 K in F-doped LaOFeAs has started a rapid and intense search for new Fe-based superconductors. After the initial class of compounds with chemical formula REOFeAs (RE=Rare earth) - also called 1111 in literature - with a maximum Tc=55 K in SmOFeAs, superconductivity has also been observed in 122 compounds (AEFe2As2), 111 (AFeAs) and 11 (FeSe1-x).
All these structures contain square planes of Fe atoms, surrounded by distorted tetrahedra of As(Se). Superconductivity seems to be unconventional because, like in the high-Tc superconducting cuprates, the electron-phonon interaction calculated by LDA is extremely small, and the phase diagram shows that superconductivity occurs close to a magnetic instability.
At the moment, not only the superconductivity, but also the basic electronic structure of these materials is not completely understood. The field is rapidly evolving, but a nice review can be found here.
Using linear response we have have first shown that superconductivity in Fe-based superconductors cannot be explained in terms of Migdal-Eliashberg Theory of superconductivity. We have then shown that, even though Density functional calculations seem to capture the essential features of magnetism in this class of compounds, the quantitative values of magnetic moments and stabilization energies are strongly overestimated with respect to the experiment, and are extremely sensitive to the computational details of the calculations.
Lately, we have been working on an ab-initio Tight-Binding model of the electronic structure of LaOFeAs, which should be easily extendable to Fe-As superconductors, and we have so far applied it to understand the magnetism.

My papers on the subject:

L. Boeri, O. V. Dolgov, and A. A. Golubov, "Is LaFeAsO1-xFx an Electron-Phonon Superconductor?, Phys. Rev. Lett. 101, 026403 (2008).
I. I. Mazin, M. D. Johannes, L. Boeri, K. Koepernik, and D. J. Singh, "Problems with reconciling density functional theory calculations with experiment in ferropnictides" , Phys. Rev. B 78, 085104 (2008).