Anybody have any idea why a computer would randomly reboot itself? My computer keeps doing it, I don't know why. Some times it goes for days without doing it, other days it goes any from 15 minutes to an hour before auto-rebooting.
I'm running Windows-ME on a Intel Celeron 700 Mhz.
Help! 
Have you actually been looking at the screen when it would randomly reboot? Where there any error messages? Any blue screens? What were you doing? Does Windows come back up complaining about an unsafe shutdown?
In general, what applications are you running? Anything in common with each time it reboots?
Try running a full system scan using your antivirus software with the latest signatures. No antivirus software? Try
AVG Anti-Virus. It's free.
Along that vain, make sure you're not running any wacky spyware. Use
SpyBot to do a system scan. Make sure you download any updates first.
How's your hard drive space? Do you have plenty (500MB+) free?
How old is the computer? What kind of computer is it? Is it under warranty still? Call the manufacturer.
If it's a homebrew system, try swapping out the memory for new or other known-working modules.
Someone else mentioned overheating. If you have any reletively recent BIOS, it will have the option to report CPU temperature. Download
SpeedFan and see if your CPU is overheating and halting your system. Usually, it will just halt the system and not reboot it, but who knows?
I also had a problem where my system would seem to reboot itself randomly and it was directly related to the video card + video card driver combination I was using. It seemed to happen when I was doing any windowing function, like opening, closing or moving a window. Update or downgrade the video card drivers and see if the problem goes away.
As Howard_Casto articulated, Windows ME isn't exactly the best OS in the world when stability is a factor. I would suggest either upgrading to Windows XP, Windows 2000, or even downgrading to Windows 98SE (Yes, Windows ME is _that_ bad!).
Good luck. Let us know how it works out!
/Steve