One of the most obvious things about the Moon is that it is covered with impact craters, but these are not uniform. Some older areas called the highlands are very heavily cratered where as the 'mares', thought to be seas by early astronomers, have much fewer craters. This is because the mares have been resurfaced with molten rock more recently showing that the number of impacts has reduced over time.
One theory about these craters says that there was an event called the Late Heavy Bombardment about 3.8 billion years ago which was both far more intense and different from the impacts that have occurred since.
As going to and dating each individual crater is ridiculously expensive, you have to study thousands of craters from orbit and use statistics to date them. Normally this is done from photographs, which is a problem, as different lighting, can change which craters are visible, causing problems for your statistics. Howver the Lunar reconnaissance orbiter which was launched last year includes a laser altimeter which has built up a 3d map of the moon avoiding these complicating factors.
James Head and collegues have been studying craters larger than 20km across using this map. One result was that the lunar highlands the craters cover 3-10% of the landscape, which is effectively the highest you can ever find, as new craters erase the older ones, so there is unlikely to be any original lunar surface left.
They have also been comparing different aged surfaces. The younger ones have the distribution of crater sizes you would expect from the asteroids which are presently crossing the earth's orbit. However the older highlands dating from before the Late Heavy Bombardment seem to include far more large craters then they should, and in fact look very like the distribution of asteroids you find in the main belt of asteroids.
This indicates that the Late Heavy Bombardment wasn't just a more intense version of the bombardment that has occurred ever since, but something different. One theory is that the orbits of Jupiter and Saturn changed during the early solar system setting up resonances in the main asteroid belt destabilising it, and hurling a random selection of its contents into the inner solar system, which would of course have had an even more catastrophic effect on the larger and heavier Earth possibly having huge effects on Earth life developing at this time.