[I]n 2007 the Gravity Probe B team confirmed one prediction of general relativity. According to Einstein, the Earth’s gravity warps spacetime like a bowling ball on a trampoline. This geodetic effect was measured with an error of about 1 percent. [The theoretical prediction for the geodetic effect in this experiment from general relativity was 6.61 arcseconds (1.84x10-3 degrees) per year.]
The much-smaller frame-dragging effect from the Earth’s rotation, though, remained hidden in the noisy data. Theory predicted frame-dragging should change the orientation of the spinning spheres by only 39 milliarcseconds per year, about the width of a human hair seen from 400 meters.
After NASA pulled the plug in 2008, private funding arranged by an executive at Capital One Financial and the royal family of Saudi Arabia bought some extra time to clean up the data. . . . The results of this painstaking analysis, scheduled for publication in an upcoming Physical Review Letters, reconfirm the geodetic effect with an error of about 0.2 percent [about 12 milliarcseconds]. Gravity Probe B puts the frame-dragging effect at 37 milliarcseconds with an error of about 19 percent [about 6 milliarcseconds], far from the original goal of 1 percent precision.
From Science News.
An experiment with the LAGEOS satellites whose results were published in Nature in 2004 had already confirmed the frame dragging effect with an error of 10 percent, but the independent replication of this result still has value. The team that achieved that result hope to improve their precision to 1% with a satellite launched this year.
Suffice it to say that the results are accurate enough, and the track record of general relativity's predictions in this domain (weak field, low speed), are good enough that nobody is betting that these predictions will be disproved. The experimental results are within 0.4 sigma of the theoretical prediction, well within the range of what would be expected from experimental error and not a result of cherry picking given that the result has been experimentally tested only twice.
The dark energy effects which have been observed to date can be incorporated completely into the equations of general relativity simply by setting the appropriate value for the cosmological constant, although there are alternative explanations as well. The uncertainty in this constant is currently about 3.3%.
The gravitational constant "G" remains one of the least accurately determined fundamental constants of physics, primarily because it is hard to precisely measure the gravitational effects of masses amenable to laboratory measurement since the gravitational impact of laboratory sized masses is so weak. It is 6.67428 x 10^-11 Newtons (meters/kilogram)^2
with a relative standard uncertainty 1 part in ten thousand. This latest estimate from 2007 is an improvement by a factor of roughly one hundred over an error factor of about 1.3% as of 1798.
Gravitational time dilation effects have been measured directly with atomic clocks in Colorado within the last year and have been precisely as predicted by General Relativity.
Some of the missing pieces of an experimental confirmation of all aspects of general relativity include the direct observation of gravitational waves, although there are strong indirect indications that they exist as predicted.
More seriously, there is not a consensus explanation for "cosmological inflation" (i.e. much faster than cosmological constant rate expansion of the universe during the time from 10^−36 seconds after the Big Bang to sometime between 10^−33 and 10^−32 seconds after the Big Bang). Still, a glitch this tiny, for that short of a time period, 14 billion or so years ago, isn't exactly a glaring flaw. The Big Bang theory still provides a simple, consistent explanation from that time forward, and there are all sorts of uncertainties, some of which may never be possible to resolve, about the extremely early cosmology of the universe.
Also, there is some reason to believe that singularities aren't as singular as they appear in classical general relativity equations at a quantum level.
The only observations from astronomy that do not fit the equations of general relativity, which was proposed in 1916 by Albert Einstein, and the directly observed distributions of matter, within the realm of experimental accuracy, are those attributed to dark matter, most commonly via a cold dark matter paradigm.
The main problem with this theory is that a dark matter particle that would be a fit for the model has never been observed directly. Indeed, experimental results about the potential mass of such a particle so far this year are contradictory. DAMA and COGENT experiments seem to show annual variation in the frequency of events attributed to a dark matter particle proposal in the mass range of about 7 GeV/c^2, something seemingly ruled out by other dark matter detection experiments, one of which used the same material in its detector as COGENT.
Another problem is that a naive cold dark matter model do not fit the observed data, because that model produces the wrong dark matter distribution to predict the effects expected (cuspy halos), an insufficient number of dwarf galaxies, and incorrect amounts of angular momentum in galaxies. Efforts to determine the amount of ordinary matter in the universe also make estimates of the right amount of dark matter error prone. There was a major underestimate of the amount of ordinary matter in elliptical galaxies that was just discovered in the last year, and there is still little clarity regarding the aggregate mass of neutrinos in the universe, because the average mass of an individual neutrino isn't very precisely established. Various efforts have been made to solve these problems, but no one consensus resolution has solved all of them yet.
According to one person I've heard explain why gravitational effects of dark matter can't be observed at the solar system level: “The density of the solar system is much bigger than the density of our galaxy, and dark matter gives 1-5 times the density of our galaxy. Utterly irrelevant in the solar system[.]”
Alternatives to dark matter such as Modified Newtonian Gravity aka MOND (TeVeS in a relativistic variant) have been proposed and work at the galactic scale, but still need some dark matter of some kind to explain how galactic clusters behave (possibly simply massive neutrinos, however), and have not provided a good explanation for the behavior of the "bullet cluster" collision. But, efforts to validate this hypothesis have shown that a simple equation with just a couple of constants (that calls for gravity to fall off at 1/r rather than 1/r^2 below a critical value for gravitational field strength) can accurately predict all dark matter driven phenomena at the galactic scale in all types of galaxies (including some whose behavior was predicted before it was measured), so any dark matter theory that works must accurately reproduce this relationship at this scale.
The bullet cluster observation also places the tighest limits to date on the cross-section of interaction of dark matter, which in inextricably intertwined in experimental practice in most cases with dark matter particle mass. This isn't wonderful for the cold dark matter model because it "rules out most of the [cross-section of interaction constant] range invoked to explain inconsistencies between the standard collisionless cold dark matter model and observations."
One of the main effects of the experimental confirmation provided by Gravity B and other general relativity testing experiments has been to tightly constrain the extent to which any quantum gravity theory that does not reduce exactly to general relativity can deviate from it.