Several privately run organizations, aiming towards building a Lunar settlement, have been launched in the past. Some have been around for a while, such as Moon Society and Lunar Explorers. Some have just recently emerged, such as Lunar Union, and generally seem to have a more think-outside-the-box approach. Then there are the government run organizations, such as NASA, who have plans for setting up a base on the Moon. Their agendas differ and so do their means and strategies for reaching their goals.
We can only speculate in what organization will succeed and to what extent. Due to the funding available to government organizations and their small scale objectives, one can assume that one of those will be first (e.g. NASA's Lunar Outpost). But a single base hardly constitutes Lunar colonization. Colonization really is about many people working together, and in that respect Lunar Union seems to be spot on.