Overpopulation won't be a problem -- one day. As populations age they slow down their growth, until, it seems, they actually start to shrink. We don't know if this will hold true for every country, but it is happening in Europe, North America, and Asia. This is largely because educated women who are able to take control of their reproduction will on average have fewer children.
Population growth by country:
These numbers are from the CIA World Factbook, updated 2009. And yes, those minuses do mean a negative population growth. The USA and the UK are higher than other developed countries largely because they have so many immigrants from less developed countries, and these immigrants have higher birthrates than the WASPs Yuppies. Much of Asia has a growth rate of less than 1%, including China (at 0.655%). All of these countries (and many others besides) are showing a general trend of decrease in the rate of population growth, and it is predicted that in the next few decades the only thing keeping them from negative growth might be immigration.
Of course, there are many countries that have people starving, dying of malnutrition and poor water supply, or being killed over things that are in short supply because of overpopulation and poor resource management. The developed nations pump billions of dollars of aid into these countries, and they are making some headway. You can help; there are many organizations that would be happy to take your money and give it to someone in need. (Ad for PATH goes here).
However, you, personally, are unlikely to see a world in which there are no countries with overpopulation. Your grandchildren will likely live to see such a world, although if that world has flooded or had the oceans poisoned there will surely be starvation for many decades more, as the excess billions die off. Eventually, though, we will reach sustainable population levels.
As a sidenote, world population is measured in billions
, so sending people into space colonies is not likely to become a useful strategy. Sending one billion people to Mars will likely remain unfeasible. Sending one thousand people and waiting for them to multiply is a much more reliable (and affordable) method of settling the universe.