The final parsec "problem" Many and perhaps all galaxies seem to contain supermassive black holes of about $10^7 M_\odot$ at their centres. Determining their origins is of great astrophysical interest. 
In what I understand to be the currently-favoured theory of "hierarchical merger", the supermassive black holes get larger over time via repeated mergers between smaller holes, during for example galactic mergers. This requires a natural mechanism dissipating orbital energy and angular momentum between the holes such that they will generically merge in much less than a Hubble time.
When the holes are very far apart (much further than a parsec) such a mechanism exists in the form of gas drag (EDIT: the actual mechanism here is dynamical friction; thanks Chris White for pointing this out). And when they are very close the gravitational radiation reaction becomes important. But at distances of about a parsec the holes don't seem to be driven together, and according to current theory should never merge. Thus the "final parsec problem": how do we get the holes close enough together for a merger to happen?
However as far as I'm aware there isn't actually any evidence that the black holes ever do merge. A large swarm of small black holes about a parsec apart would be equally compatible with observation. Or else some process other than hierarchical merger could account for the holes. So it seems this "problem" has more to do with the careers of gravitational wave scientists than with any actual difficulty in astrophysics. Or am I being unfair? 
 A: Great question.
1) There is indirect (and circumstantial) evidence that they do merge.  While there are some famous examples of apparently 'binary' (or more accurately 'dual') AGN (e.g. Komossa+2003, or Rodriguez+2006) there seems to be a very conspicuous dearth of such systems --- suggesting that they don't spend very long at observable separations.  Note that "observable" doesn't necessarily mean (optically) resolvable as VLBI has also found very few binaries, and there are also spectroscopic methods which should be more sensitive the closer the systems are.  Additionally, people have suggested that the tightness of the M-sigma relation suggests that BH binaries merge effectively.  Finally, all of the well constrained observations of BHs (e.g. through stellar velocity profiles) can fairly reliably rule out the presence of massive companions.
2) There are growing theoretical reasons to believe the 'final-parsec problem' isn't as bad as previously thought.  One of the best reviews of the subject is given by Merritt and Milosavljević 2009.  
On the scales of parsecs, the dominant 'hardening' mechanism (bringing the binary closer together) is through scattering of single-stars in the so-called 'loss-cone'.  If you take the simplest analytic model for the loss-cone, of a spherically symmetric, isotropic distribution of stars, then the loss-cone is rapidly 'depleted' --- there is only a small number of stars which are able to scatter with the binary, and once they do the loss-cone is 'empty' and very slowly refilled.  More realistic models, which take into account perturbations from anisotropic and nonspherical systems suggest that the loss-cone can be refilled much more rapidly and efficiently.
Additionally, the importance of local gas-drag on the binary on parsec and sub-parsec scales seems to be much more significant than originally thought.  This is for two reasons: i) large amounts of additional, dense gas seem to get funneled towards the galactic center during mergers.  ii) the formation of a circumbinary gaseous disk can be much more effective at extracting orbital energy / angular-momentum than was initially thought (e.g. Cuadra+2009, MacFadyen and Milosavljević 2008).
Right now, all of the theoretical models suggest that a very large fraction of MBH binaries should merge very effectively, i.e. on timescales less than about $10^8$ years.  Even pretty substantial modifications to the knobs of these models, don't seem to change the basic conclusion that we are on the verge of detecting MBH Binary systems with Pulsar Timing Arrays (PTA).  The only big caveat to this, in my opinion, is that the frequency range which PTA are most sensitive to - could actually be at the frequencies where MBH Binaries spend the least time.  None of the models I've seen, however, suggest we should be too worried about it.
All of this is an extremely active area of research, and both the observations and the theoretical models are very difficult to perform --- but hopefully we'll be seeing some exciting progress on both fronts very soon.
