Binary Black Hole Solution of General Relativity? This is rather a technical question for experts in General Relativity. An accessible link would be an accepable answer, although any additional discussion is welcome.
GR has well known solutions relating to single Black Holes: Schwarzchild, Rotating & Rotating with Charge. These solutions demonstrate some non-trivial GR behaviour. However do there exist any (corresponding) binary star/black hole solutions? Because of the non-linearity of GR such a solution could well demonstrate additional properties to a "solution" that simply consisted of a pair of "distant" Schwarzchild solutions glued together.
 A: There are no exact solutions, only approximations and numerical solutions.
Don't forget that orbiting black holes will radiate gravitational waves so any solution would have to include those and the corresponding decay of the orbit until the black holes coalesce.
A: According to general relativity, a pair of massive bodies that orbit each other emits gravitational waves - for analogous reasons to the reasons why accelerating charges in electrodynamics emit electromagnetic waves.
So there can't be any static solutions resembling binary stars or binary black holes. The solutions have to be non-static and a complicated system of two orbiting bodies that emits gravitational waves - and eventually collapses into one object - clearly can't be solved analytically.
These things are usually discussed numerically, see other answers. In particular, the 1993 Nobel physics prize was given for an observation of a pulsar whose frequency changes in time exactly in the right way to be explained as the loss of energy caused by the emission of gravitational waves as predicted and calculated by general relativity.
A: For a very recent authoritative review of the numerical approach, see Centrella et. al.  http://arxiv.org/abs/1010.5260
For the alternate parameterized post Newtonian approach, see Living Reviews of Relativity http://relativity.livingreviews.org/Articles/subject.html
and look for articles number 2007-2, 2006-4 and 2003-6.
A: Franz Pretorius has worked on this and developed animations.
http://prl.aps.org/abstract/PRL/v95/i12/e121101
The field is numerical relativity. Matthew Choptuik also, I believe, has done work on this.
A: A way of physically thinking about this is that a two body problem in general relativity does not generally have closed orbits.  If one of the bodies is very large and the other a small satellite the problem is integrable.  The periapsis (perihelion) advance of the small satellite is repeated with each orbit, which makes the problem integrable.  If the two bodies are of comparable mass the orbits of the two are perturbed in a manner which emulates a third orbiting body in Newtonian mechanics.  The three body problem is not integrable in general.  The periapsis advance of either mass adjusts to the changing position of the other mass, which “emulates” the presence of a third body.  Curiously, before Einstein people thought there was another planet near the sun which perturbed Mercury, what they called planet Vulcan.   If the two bodies are close enough and they are in an orbit with a quadrupole moment (Keplerian orbit, ellipses etc) there is the emission of gravitational waves.  Gravitational waves are mass-energy and contribute to the gravity field.  So a two body system in effect generates what might be thought of as a third body, or N-bodies.
A: The basic problem is that the holes must radiate. The result is a completely asymmetrical spacetime that cannot be attacked analytically. There are a few ways around this.
You can use the post-Newtonian expansion. Here, GR is formulated as a series of corrections to Newtonian gravity in powers of $\frac{v}{c}$. The expansion is now known to very high order and remains integrable. It seems to give quite accurate results.
You can integrate the EFEs numerically. This is possible because, while entire binary spacetimes are very difficult to find analytically, it is still possible to find families of solutions modelling a single spacelike hypersurface of such spacetimes. In principle, since GR has a formulation as an initial value problem, it is then possible to integrate forward for as long as you want. Even if the initial slice isn't especially realistic, the no hair theorems give some comfort that after a quick relaxation period during which the unphysical deformations get released as "junk radiation", the actual inspiral becomes generic. Actually performing the simulations is very difficult for a number of reasons: the first binary black hole inspiral and merger was not successfully completed until 2006. 
You can perturb around an exact black hole solution. This is the goal of the so-called "self-force" program. The idea is that the smaller black hole should deform the 'background' metric in a way analogous to the electromagnetic radiation reaction, or the QED self-energy. This turns out to be really hard to do in practice, although some progress has been made for Schwarzschild backgrounds.
You can construct a few highly unphysical solutions with multiple black holes. These typically have some kind of bizarre feature that somehow holds the holes in place. For example this paper http://iopscience.iop.org/article/10.1088/0264-9381/31/22/225009;jsessionid=CC35FAD5AE9913F094348033E0C4776D.c2.iopscience.cld.iop.org studies a pair of black holes held rigidly in place by an extremely pathological "cosmic string". Another possibility is to feed in gravitational radiation from infinity in such a way as to \emph{exactly balance} the radiation reaction, cancelling the inspiral. I'm not sure whether an exact solution exists for the latter case, but if the holes co-rotate you do get back at least one Killing vector (corresponding to a "helical" symmetry).
A: You can take a homogeneous and isotropic background, cut out a "vacuole", and place a Schwarzschild black hole inside, carefully matching the densities. Perform this multiple times and you have a "Swiss Cheese" universe. Similarly, "black hole lattices" consist of multiple black holes in an arrangement based on a Platonic solid for example.
While neither of these exact solutions is a "binary black hole" in the sense of an orbiting pair, they do combine multiple black holes in one spacetime.
