Non-Newtonian motion of the moon
The motion of the moon, as measured by the Lunar Laser Ranging data is not as predicted by Newton. As the scientist being interviewed says, Newton's model was correct to the accuracy of the data available to him, and was good enough to land the Eagle on the moon in '69. However the Lunar Laser Ranging data can position the moon to an accuracy of 3cm. 

Relativistic effects cause the moon to deviate from the orbit predicted by Newtonian mechanics by about 1m, mostly due to special relativistic effects such as Lorentz contraction. General Relativity accounts for a variation of 10cm. This is much less than the perturbation due to Jupiter (about 1km). However, general relativity does fully account for the motion of the moon, to the accuracy of the experiment.


  All tests are in agreement with Einstein's General Relativity which is used for the numerical integration of ephemerides. No variation of the gravitational constant is discernible. Source


Lunar ranging is ongoing at the Apache Point Observatory Lunar Laser-ranging Operation.