There is a very interesting article in the new issue of Nature Physics regarding the twentieth anniversary of the discovery of high temperature superconductivity. In case you've been living under a rock since 1986, the high temperature superconductors are generally based on perovskite quasi-two-dimensional compounds that have extraordinarily rich (read: so complicated they're hard to understand) phase diagrams. The parent compounds are antiferromagnetic insulators in their ground state. In doped compounds (done by substitutional chemical doping at the ten percent sort of scale, which introduces significant disorder), the superconducting state is well-described by a BCS-like state with spin singlet d-wave Cooper pairs (and just establishing that firmly took years, and an enormous effort at sample growth, and several brilliant experiments).
The normal state of these materials is a real mess. At very high doping levels, the materials seem to be well-described as Fermi liquids, which is the standard picture of ordinary metals. You can think of the electrons as partially filling a band of states that look very much like non-interacting single-particle states. Excitations above the ground state look like well-defined electron quasiparticles, as demonstrated by, e.g., a resistivity that varies like the temperature squared. Near optimal doping for the superconductivity, the normal state is a "strange metal", meaning that the resistivity varies with temperature like T, implying that quasiparticles are not a sensible way to think about excitations of this material. In underdoped materials, the normal state looks like a strange metal with a "pseudogap", vaguely reminiscent of the superconducting gap in the density of states, but persisting up to much higher temperatures than the superconducting state.
The Nature Physics article is a collection of comments by a bunch of big-name condensed matter theorists. Interestingly, and I'll write more about this in a day or two, there still is suprisingly little concensus about what's really going on in these materials. Definitely worth a read!