In the Newtonian view of the world, binary star systems should remain in a stable orbit in perpetuity, no matter how massive the objects or how close the orbit. But with general relativity, that changes; energy gets carried away from the system in the form of gravity waves, which gradually causes the orbit to decay, ultimately leading to a merger.
By observing binary systems of massive objects, we've determined that general relativity gets it right. These systems behave just as general relativity predicts, giving us confidence that the theory is correct. What's missing is the other half of the confirmation: gravity waves. We haven't detected any originating from these systems. In fact, we haven't detected any, period.
It's not for lack of trying. For nearly a decade, the Laser Interferometer Gravitational-Wave Observatory, or LIGO, searched for gravity waves from astronomical events, like the merger of two black holes (a number of other detectors have also engaged in the search, but all have come up empty). Now, scientists are readying a worldwide network of LIGO-like detectors that should start coming online in 2017. A short perspective in Science outlines the project's plans.