In the early 80's, Taylor and Hules (recent Nobel prize winners for this work) discovered a system where one pulsar circles another compact object. Because the pulsar pulses occur at very regular intervals, they can be used as a clock. Moreover there are several physical effects which can be used to determine the shape of the orbits of the pulsar and the compact object. It was found that these objects are slowly spiraling into each other, indicating that the system is losing energy in some way.
This system can also be studied using the General Theory of Relativity which predicts that the system should radiate gravitational waves carrying energy with them and producing the observed changes. These predictions are in perfect agreement with the observations. This is the first test of General Theory of Relativity using objects outside our solar system.