I have an Azden shotgun mic that goes into my camcorder via XLR. It prouduces a buzzing background noise, unless I touch the metal bit on the cable that connects it to the camera.

At the moment I need someone to stand there and touch it whilst I record so that the sound doesn't appear.

Any ideas how I can get rid of this buzz?

Check to see if another cable produces the same result. Sounds like you have a bad ground (I believe you folks in the UK call it a bad earth).

If it's a cable run, also make sure it's not laying over any power supply units and other mains power cables.

Of course you could take a couple of feet of wire and put an alligator clip on each end. Clip one end to the "metal bit" on the XLR plug and the other to your earlobe while you are shooting <grin>.

Steve: I thought alligator clip earring died a hideous death at the end of the 80's...

Definitely sounds like the cable. First thing to do is try another, as has already been suggested.

If it's not the cable, check the screws on your microphone. I had this problem once when one of the tiny screws on my microphone had fallen out. The part with the XLR plug was loose and creating electrical havoc.

Some Azden shotguns from the past weren't balanced even though they had an XLR connector. That's one more variable to consider when you test a different cable, but hopefully just switching to a new cable will solve it with no further head-ache (or ear-ache).
Which camera? Which Azden shotgun? Are you operating the camera on AC or battery power?