Can the denominator of a fraction ever be zero?

I've heard this discussed a few times recently and people who seem intelligent have been advocating for the affirmative opinion. I don't understand how dividing anything by Zero is doing anything at all. Isn't One divided by nothing still One?

Edit: So no one seems to agree on this question. The answer is either 1,0, undefined or infinity. Gentlemen I think we have a paradox. This reminds me of the quantum physics versus classical physics problem or Schrodinger's theory. If the answer of the question is undefined then does that mean we just don't know mathematics at all or are the fundamental laws of the Universe broken and we live in a flaw universe which is doomed to fail?