|MadSci Network: Astronomy|
Yes, the Moon is receding from the Earth slowly. The reasons why have to do with tides and conservation of energy and angular momentum, and are about a college-junior-level homework problem to work out.
There is direct observational evidence for this. The Apollo 11, 14, and 15 astronauts left retroreflectors on the Moon, and the Soviets put retroreflectors on two of their unmanned lunar landers, just for this purpose. You can shoot a laser pulse through a telescope at one of these retroreflectors and some of the beam will be reflected back to the telescope. (The beam spreads out during the trip, 384,000 kilometers each way, so not all that much actually makes it back; you have to do this for a number of minutes to be sure that you got a good amount of measurements.) We know the speed of light with great accuracy, so if you measure how long it takes between your laser shot and the time the pulse returns, you get a very precise measurement of the distance between your telescope and the retroreflector on the Moon. A single good measurement gives an accuracy of about an inch. Measurements have been collected now for over 25 years.
The Moon's orbit around the Earth isn't a circle, so over the course of one month it moves towards and away from the Earth by a few per cent. This means a small number of measurements don't suffice to tell what's going on. If you track the Moon over time, though, it is clear that the Moon's orbit is slowly growing larger and that the Moon is moving away from the Earth. The net result is that the Moon is receding from the Earth at about 4 centimeters a year.
(Incidentally, this isn't the only evidence for recession of the Moon from the Earth. The ancients observed and recorded solar eclipses pretty well, and those observations don't match what one would expect based purely on unchanging Twentieth Century orbital parameters for the Moon.)
We aren't limited to doing this laser ranging trick to retroreflectors on the Moon. Several satellites have been equipped with these also and laser ranging of artificial satellites has been going on for a long time as well.
There are short descriptions of this science here and here. You can also try typing the phrase "lunar laser ranging" into your favorite Web search engine, though many of the items this returns will be technical reports long on discussion of methods and measurement programs and short on results.
Try the links in the MadSci Library for more information on Astronomy.