Theseus' Paradox: Are You Still You?

Theseus is known as a legendary ancient Greek king who is the supposed founder of Athens.  He appears in many myths and legends, but is most famous for his ship: the Ship of Theseus.  It was a grand ship that was gifted to him by the people of Athens and preserved as a memorial to him for hundreds of years.  Over time, however, one of the ship's planks began to rot.  So the plank was promptly replaced.  A little while later, another plank began to rot, so that plank was replaced as well.  Eventually, after years and years and years of this, none of the planks were the same.  Each one had been replaced at some point or another.  The question: is it still the same ship?

If your answer is no, then at what point does it stop being the same ship?  Once the first plank is replaced?  The last?  Halfway through?

If your answer is yes, then why?  If the ship had been taken apart and put back together with completely different pieces all at once, then would it still be the same ship?  Or does it only apply when the change is gradual?


Now, consider this:
Cells in your body do the same thing the planks do.  They die out, and they're replaced.  After about seven years, each cell in your body has been replaced and none of the original cells remain.  


Are you still the same person?  If so, why?