The Future of Humanity Institute has released a technical report on the feasability of whole brain emulation (PDF) — reproducing the brain, and hopefully the mind, in a computer medium.
The basic idea is to take a particular brain, scan its structure in detail, and construct a software model of it that is so faithful to the original, that when run on appropriate hardware, it will behave in essentially the same way as the original.
On the one hand, co-authors Anders Sandberg and Nick Bostrom do an excellent job of outlining just how speculative WBE still is and the plethora of breakthroughs and advances that would be required just to move backing up your brain from scifi idea to even remotely possible.
Computing power turns out to be one of the likely limiting factors as at present the processing power need to handle all of the neurons is insanely large. On the other hand, as Sandberg and Bostrom note we already know of at least one machine that is perfectly cable of running such enormously large parallel computations — the brain itself — so we already know it is possible. We just have to figure out an efficient way to reverse engineer the damn thing.