Who has the burden?

Here's something that happened again to me today; I was told that I have the burden of proof to say how UFOs aren't real. I tried to explain in the Q&A session how I did not have a burden of proof, yet the opposition kept trying to make the point. Of course, the judges picked up on this, and gave me the round. Huzzah. But these kinds of things are becoming all too common. So, I will try and explain how the burden of proof and the burden of rejoinder works.