Present Day, Future Plans
Well, so we haven't managed a lunar base quite yet, but what are the plans for the future?
The US has the most established program to get ourselves back on the Moon—and this time, permanently. The Vision For Space Exploration mandate for NASA, established in 2004 by George W. Bush, calls for human exploration of the Moon by the year 2020. The idea is that if there is a base on the Moon, then space exploration in general could be made much easier: mining the fuel (helium 3) on the Moon, launching the spacecraft from lower gravity, and manufacturing water and air from the soil, among other advantages. A base would probably be located at one of the poles so that it can have continuous solar power, a place which is also more likely to have lunar ice to be used for water, which can be split into hydrogen and oxygen, all important for survival.
So far, the States has a pretty good foothold. The recently launched LRO mission is a beginning to this. While unmanned, the mission focuses on gathering the data NASA scientists need to plan lunar bases out in more detail, especially with regards to potentially water and air.
Of course, the United States is not alone. The European Space Agency is hoping to have a lunar base established by 2025, and Russia a manned mission by the same year. China's also got it's eyes on the Moon. They've estimated that by 2022, they'll have the technological capability to do so—and they've certainly got the will for it, already ambitiously planning for lunar mining projects for helium 3 and iron. Japan and India's current plans involve no permanent residents on the Moon, rather, a rotating series of crew members similar to the International Space Station.
Who's going to have a lunar base first? No one is entirely sure, though most people are still betting on the Americans. What is certain, however, is that the race is on.