The gentleman giving the speech in the embedded video is Aubrey de Grey, and he is giving this speech at TED. The point he is making is about immortality, and that basically it is achievable.
Obviously, many people's first reaction to this last sentence is likely outright denial - that this is impossible. You could think that he is probably mad, or highly delusional. However, if you do take the time to watch this speech, you will recognize that de Grey is very much sane, and in fact, quite intellectually and convincingly arguing that human life can be extended indefinitely. He further puts forward quite rationally that it is not just possible, but also should be achieved.
I would suggest everyone to watch this video - if only to get a different perspective. One of the key points in the speech is the graph at the point 10:45. What he basically says is that the longer we can keep any individual alive, the more chance the individual has of living to a time when his age will be extended even more. This, inevitably leads to a conclusion: We might never cure all health problems related to ageing and death, but, we can solve them fast enough that nobody needs to die.
That is an intriguing thought. Normally, we would imagine that although longevity is obviously possible, we would imagine "immortality" to be impossible, because no matter how many cures we have, or how much advancement we make, we won't be able to solve everything. But, de Grey makes a different claim - he says that we do not need to "solve all the problems", but only to "solve them at a faster rate than the people dying."
Unfortunately, as convincing as this may be, I respectfully disagree. You see, de Grey's suggestion has a hidden underlying assumption - that the discipline of medicine can grow at a rate to catch up with the health problems we will face. So the graph is valid only if you assume that the rate of advancement of medicine to beat the rate of problems in a human's lifetime. I will now argue that this is not only unlikely, but impossible.
Medicine, like any other science, advances by experiments. Inevitably, the subjects of the experiments of medicine are humans. Even new medication requires decades to develop. This is part of the nature of medicine. No matter how many guinea pigs die, the doctors know that a certain cure will work only after it has been tested on humans. The doctors will need to come across the problems, work on corpses, and watch how the diseases progress as their patients die to be able to understand the nature of the disease. Therefore, medicine relies on experiment, on humans, and requires the recycling of human generations. Therefore, the rate of advancement can never be high enough to beat Death indefinitely.
That being said, there might be breakthroughs. Our two most serious problems are cancer, and heart problems. If these two problems can be overcome, we basically don't have a next cause of death. For example, if we can grow hearts in labs, which actually looks possible, than it means that we have cheated the most common cause of death. If cancer could be possibly avoided, which actually looks unlikely, given the complex nature of all different types of cancers, we could extend the human life. But other diseases would spring to take their place, we just have not seen them yet.