Has Chandra Varma explained cuprate superconductivity? Chandra Varma is a theoretical physicist at University of California, Riverside. A couple years ago, he gave a talk at my institution purporting to explain superconductivity in the cuprates. It all sounded so great and convincing, and I want to know what the status is with that.
Here (link) (arxiv version link) is a 2006 paper explaining his theory in detail, and here (link) is a more recent one. Supposedly, there is a spontaneous breaking of time-reversal symmetry, resulting in microscopic currents with long-range order, running in loops between the copper and oxygen atoms. These currents can function as the glue for electron-electron pairing. These currents, which are very hard to detect, have actually been seen by spin-polarized ARPES and by spin-polarized neutron diffraction. Here (link) and here (link) is the spin-polarized neutron diffraction study, and Here (link) is a more recent 2010 neutron-scattering experiment confirming the same results, published in Nature. Here (link) is a report of him discussing his theory with other experts in superconductivity theory.
So, as far as I can tell, this is a simple, elegant, experimentally-proven theory explaining cuprate superconductivity. The theory and supporting experiments are at least five years old. But everyone still says that cuprate superconductivity is a mystery. What's the deal? Is there a problem or controversy in this theory? Does the theory explain only a small part of the mystery of superconductivity? Am I misunderstanding something?
(I am a condensed-matter physicist but not a superconductivity specialist.)
 A: OK - taking the questions one at a time.  Full disclosure: I'm a member of the phonon tribe, but I'm trying not to let that cloud my response here.
"So, as far as I can tell, this is a simple, elegant, experimentally-proven theory explaining cuprate superconductivity. The theory and supporting experiments are at least five years old. But everyone still says that cuprate superconductivity is a mystery. What's the deal? Is there a problem or controversy in this theory?"
Sure: So why does [your field] need a whole journal, anyway?
Joking aside, yes, there are problems with this theory. From the experimental side, for the most recent evidence, the presence of orbital currents was excluded in the pseudo gap phase using NQR[dx.doi.org/
10.1103/PhysRevLett.106.097003
], and a muon spectroscopy study done on the very same sample that was used for the first polarized diffraction study shows that there is a substantial amount of magnetic impurity phase [dx.doi.org/10.1103/PhysRevLett.103.167002], making the clear cut interpretation of orbital currents ambiguous.  The predicted fluctuations indicate one should see a strong quasi elastic response in bulk probes like neutron scattering upon entry into the pseudo gap phase, but as far as I know, there is no evidence to this point.  This theory is seemingly absent an explanation for the observations from neutron scattering  about spin resonances (gapped excitations that are observed at the AF ordering vector).  It may be a red herring, but as the resonance is universal in unconventional SC and scales with Tc, some explanation is warranted. Further, this theory is silent on the matter of electron doping, which also induces SC in some copper-oxide materials.
Does the theory explain only a small part of the mystery of superconductivity? 
It provides correct order of magnitude estimates for the Tc and size of the superconducting gap.  It explains some of the exotic behavior exhibited in the pseudogap phase.
Am I misunderstanding something? 
Probably.  There are hundreds (thousands?) of theories of HTSC, all of different merit, and all interpreting the evidence very carefully.  As is pointed out in the comments, these things are pretty contentious, and I'll reiterate that the evidence has been refuted in several cases. 
Now, none of this answers your primary question, which is whether or not this theory solves HTSC.  The answer to that question is definitely maybe.  It does get the order of Tc and the energy gap correct, and has supporting evidence.  However, the data that these papers provide as evidence present some ambiguities in the interpretation of the experimental result in light of sample preparation issues and choice of probe.  Fundamentally, a theory of high Tc must have some predictive power of the Tc, the energy gap, be supported by most of the experimental evidence, and to some extent, what the sample makers could put into a crucible and have pop out as HTSC.  It is not clear that this theory has done the last of these two sufficiently, and some reconciliation needs to be done. 
