NASA is mapping the moon’s gravity
NASA has a mission planned which involves sending a pair of satellites to orbit the moon and take measurements of the gravity field. The Beeb has a basic explanation here. The basic idea is that the lead satellite will accelerate or decelerate in different patches of gravity, which will be detected by the other satellite as a change in the distance between them.
Why does the moon’s gravity vary? Because it isn’t homogenous, i.e. it’s mass isn’t uniformly distributed. Neither is the Earth’s, in fact – gravity on this planet varies according to latitude, the local geology and altitude. At the Equator, the apparent gravity is 0.3% lower than the actual value, because the outwards centripetal force at the equator balances out the actual gravity by that much.
Gravity does vary with height as demonstrated by Newton’s Law of Gravitation (which is a good enough approximation at speeds that come nowhere near the , but a common misconception is that astronauts in the recently-retired Space Shuttle or on the ISS are “weightless” because they are above the Earth’s gravity. This is false because at an altitude of 400km (the Shuttle’s typical orbit height), the gravitational attraction experienced by the Shuttle and it’s crew is 90% of the ground-level attraction. Weightlessness actually occurs because they’re in free-fall. If you want to see how much it can vary around the world, here is a simulation on Wikipedia. (I can’t figure out how to embed it, but I’ll try).