G5 iMAC has started to request that I shut down and restart

Welcome to the forums!

Sounds like a Kernel Panic. Have you recently added more RAM?

What is a Kernel Panic?

http://docs.info.apple.com/article.html?artnum=106227

First, you could run Rember to check the RAM.
http://www.kelleycomputing.net/rember/

And from the "Additional Software & Apple Hardware Test" disc, try running the Apple Hardware Test and see what the results are.

I just experienced the same problem with my Rev A iMac. After testing the memmory I found that it was bad. The computer starts up with no problem when I use the old memmory that came with it. Unfortunately this is not enough memmory to run Leopard. So now I wait for the new memmory to come. But this looks like it will solve the problem.

Ok. The memory came today, I have loaded it in the machine and it has been running for two hours with no problem. I think this has done the trick. At least for me!! I may get another year out of this machine yet.

Well I'm still experiencing problems.

Have you cleaned the intake and exhaust vents lately? If you haven't done so within the last six months or so you may find them pretty well clogged.

Yes that was the first thing that I did. I hit one with too much compressed air and blew it to pieces. So I ended up getting a new fan as well. I've been running the Apple hardware test that came with my iMac and sometimes it runs fine. Other times like just now it stops and I get error messages. It's reporting an script error 25 (not a struct or union element) in file #0: scripts\AHT_script.sc, line# 1234

Other errors are error 23 missing ':' in file #0:scripts\AHT-script.sc, line # 12422

Or invalid memory access at #SRR0: 00000000.06610684#SRRI:10000000.00003030

Any idea what this is all about?