Longevity is inevitable and should be encouraged. Social mores will change as the average lifespan expands beyond the 80ish years we currently experience. I have no doubts that feelings towards age of having children, use/value of accumulation of money/wealth, inherent value of educating oneself, etc., will drastically change from what we believe now. I firmly believe that true immortality will be achieved, and that it will do so with a few steps (incrementation will likely occur, but these two are the focus as I conceive of the process)
Tissue Regeneration
Current emerging biological techniques/technologies allow us to do something never done before: autogenous tissue transplantation. This provides a means to harvest a cell of specific tissue type (liver, kidney, vascular, neural etc.) expose it to a specified sequential media process and in the presence of denucleated scaffold matrices (cartilege for example) grow entirely intact organs that can be transplanted into the donor of the originating cell(s). This eliminates graft vs. host rejection, and also makes capable zero day old organs that can be repeatedly used to exchange with aging or diseased tissues.
Transfer of Consciousness
Neuroanatomy and neurochemistry will continue to evolve to the point that a digital map can be created of the human brain specific to the individual. This will allow the transfer of consciousness as discussed in the first two pages of postings on this thread. The downloaded consciousness could be into a non biological organism (NBO as described by Ray Kurzweil in the Age of Spiritual Machines) such as an android (think of a terminator with your consciousness in it).
Immortality will thus be achieved since there is now nothing that can die in a non biological entity. So long as the data image that is your consciousness does not get corrupted, you will not 'die'. Also, you simply 'save' your new digital map at a fixed interval (or even in real time) then you can 'load' yourself were anything detrimental to happen to your 'terminator'. Aspects of this line of reasoning have been utilized in science fiction shows. The best, in my opinion, was on Battlestar Galactica where the cylons' consciousness was beamed to the resurrection ships upon their death. I realize the implications that this lifestyle would have on our understanding of the value and preciousness of life. However, I think it inevitable that we will evolve to continuously change our definition of the meaning of life. It is also an inherent truth that there are those among us now that will refuse to ever adopt this altered form of life for themselves. That will cause the formation of a schism between the non biological humans and the traditional biological humans. However there are obvious advantages to the former and it will outpace the latter in the long run (as it should given that it is still the basic adherence to Darwinian evolution) leaving no remaining biological humans in existence. Weather one wishes to argue this will be the end of all human kind is debatable, but in my opinion humans are not humans because we have carbon based cells composed of 23 pairs of chromosomes. Rather, humans are humans because of the ability to form a consciousness that works collectively with other entities able to form consciousness. The carbon tissues are simply the shell that house this and those can be interchangeable without altering the core.
Sidebar Argument
Perception is reality, so your consciousness is you. The 'Ship of Theseus' argument indicated earlier in the thread simply does not apply as consciousness lies outside a tangible item that is/can be recycled. Where my reasoning obviously fails is depicted nicely in 'The Sixth Day' when Tony Goldwyn's character is cloned and awakes before the template has died. This caused a conflict of identity issue to emerge as the template still felt himself to be real and the only one while the copy also felt himself to be real and the only one. So to avoid that paradox, you simply anesthetize the template (biological you) digitally map the brain, and transfer as a file to a hard drive. Then put the template into stasis or simply destroy them. Then download a copy of the consciousness to the NBO (which for ease of transition can be made to look identically like the biological version). You will awaken no different than you do every morning and will experience the world as 'you' with no feeling of not being you since perception is reality. "Blade Runner" shows how an entity exists completely oblivious to their non-human existence. While I disagree with doing this without the consent and awareness of the individual, it does show how easily one could be convinced they are simply who they perceive themselves to be.
Additional
I also think it is quite possible that we will achieve immortality in a matrix like construct. We all forfeit our biological entities and live our consciousness inside a 'matrix' system. We would all be cognisant of how we got there and who we were. However life would become less independent and more of an amalgamation of conscious thought; something of a soup of philosophical pondering. The 'individual' could at any time leave this existence to pursue a more tangible sensory existence outside the matrix, but this would all be dependent upon the location of that existence (Earth, or elsewhere in space etc.)