The Moon moves away from Earth at a rate of 3.8 cm per year. We know this because the astronauts and two Russian ships were able to reach the moon missions of Apollos 11, 14 and 15 plus the Lunokhod 1 and 2 unmanned ships left five reflectors on the lunar surface.
From Earth, we launched a laser pulse to these mirrors that bounce back and return to Earth. Accuracy is 1 mm.
The Moon moves away because it accelerates in its orbit due to effects caused on Earth's oceans. For the same reason, the days are getting longer and longer.