For countless centuries, people have attempted to become immortal. Now, we are ever closer to this ancient goal. This is not just like the progress in raising life expectancy that has been continuing since the Industrial Revolution. It is extending the lifespan beyond what is naturally possible, about 120 years (see http://news.bbc.co.uk/2/hi/uk_news/4003063.stm , from 2004, but still relevant). New medical discoveries and technologies that are leading us to this goal are being developed faster than ever. (http://www.physorg.com/news143392265.html)
Many methods using modern science have been proposed to make people live unnaturally long. Those include genetic engineering, adding physical enhancements which will basically make people cyborgs, cryogenics, and uploading a person's mind onto a computer and abandoning the human body altogether. For now, at least, this is basically science fiction, but the goal is coming closer to realization. Many researchers are working on the scientific problem, and bioethicists are arguing over whether it is actually desirable.
Arguments in support of life extension basically revolve around the fact that they believe death is avoidable and thus is an unnecessary tragedy (see http://www.nickbostrom.com/fable/dragon.html for an explanation of this view in allegory form). Opponents of artificial life extension argue that it would cause overpopulation, drain the world's resources, and very likely be impossible or improbable in the foreseeable future (see http://news.bbc.co.uk/2/hi/uk_news/4059549.stm). Many private foundations fund research into life extension currently. While this does not seem to be a pressing issue right now, in the future it will come more into the spotlight, along with other speculative technologies such as artificial intelligence and space colonization. What do you think about life extension?
See the links on the Wikipedia page.