A few interesting things I learned:
Gravity bends spacetime which in turn bends light (we can see 2 pictures of the same galaxy due to this effect).
Newton's laws of gravity are a good approximation. but they're not fully correct. By measuring the distance of the moon from the earth, we know it is not quite in the location newton's gravity laws predict.
This is separately from the fact that the moon is actually spinning away from the earth as a counter effect for the energy it puts into our tides, energy that is lost by the moon spinning away from us.

Turns out we've also known for around a century that Mercury doesn't follow the orbit it should either and we now know that it's because of the effects of the sun bending spacetime and in turn Mercury's orbit. I also didn't know that before Einstein gave us the theory of relativity and spacetime, scientists made up a small planet not visible to us and that they called Vulcan to make up for the real orbit of Mercury (see encyclopedia britanica and Shapiro time delay).

By the way, the moon spinning away from the earth and the friction from the tides it causes on it, also causes the earth to slow down, making our days longer. Brian Cox says in one of his shows that the earth day used to be 22 hours 600 million years ago (the earth is 5 billion years old as a reminder). In another Nova/PBS show, I remember they found fosil evidence that the earth year was some 420 days (based on layer deposits), which in turn showed that the earth day was only 18 hours back then. And if you know a few things about the moon, you also know that the moon is always facing us from the same direction, so since the earth day is slowing down, in turn we are slowing down the rotation of the moon on its own axis.

Cool stuff...

If you're interested, you can watch two old Horizon (BBC NOVA equivalent) shows with Brian Cox: