Where do I start

[#1]

My computer keeps shutting down and then rebooting.
Where do I start in trying to track down the problem ?
I Have swapped the power supply so assume it's not that.

[#2]

Does this happen as soon as you switch on/boot up for first use?  or does it happen "after a while" ?

[#3]

Check the cpu heatsink isn't blocked and full of dust and that the fan works.
Boot with a Linux live CD (or Hirens XP / Win 7 etc) - see if the problem still occurs
If the issue stops with a live CD, you may have a driver issue or corrupted operating system, bung an old spare hard drive in, reinstall and see what happens. If it carries on misbehaving then probably a hardware fault.
If the issue keeps happening with a live CD then try a memory test for a few hours (on most of the Live CD's)

[#4]

If you can get into the BIOS screen then you may find some information on temperatures, voltages etc. or 'health' data (some give loads of info, some give very little).

[#5]

Don't know if this helps, but my sister had a dell laptop a couple of years ago and kept doing something similar. They told her to take the battery out and run it through the Ac adapter and it stopped happening. I just can't remember when it went back to Dell  whether they replaced the battery or the connection inside the laptop!  It even happened when the battery was in the laptop but she was running of mains power!

[#6]

What is the make / model computer
Disable Auto Restart and you might get a blue screen with error codes.

Prevent Automatic Restart after BSOD.
In Windows XP.
Click on Start and then on Control Panel.
Control Panel double click to open System.
System Properties window click on the Advanced tab.
In the section labelled ‘Startup and Recovery’, click on the Settings button.
In the section labelled ‘System failure’, remove the checkmark from ‘Automatically restart’.
Click OK, OK to exit.
Vista & 7
Click on Start and then on Control Panel.
In the Control Panel double click to open System.
In the left panel under ‘Tasks’ click on Advanced system settings.
In the section labelled ‘Startup and Recovery’, click on the Settings button.
In the section labelled ‘System failure, remove the checkmark from ‘Automatically restart’.
Click Ok, OK to exit.

[#7]

It's gone completely now, no power, no nothing.
Am in the process of checking out the power supply, seems that to do this you need to short out the green wire with the black wire on the 20 or 24 pin motherboard connector. followed by checking the leads with a multimeter.
If it does turn out to be a duff power supply and I purchase another one is there any possible chance that something that has shorted out on the motherboard or somewhere else could damage a power supply ?
Looking at feedback from other who have purchased power supplies seems that I have been fairly lucky not to have had a problem before now.

[#8]

What make / model / rating is the PSU ?

[#9]

650 Watts.
It was a pre assembled computer.
Have taken the 20 pin socket and bridged the wires mentioned in the previous post but left everything else connected, outputs are showing as ok +5v so reconnected everything and switched on but with the side of the case off, kept attempting to fire up with both cpu fan & internal fan going for a few seconds and then shutting down for a few seconds and then starting up again. Seems odd as it was completely dead before.
Have cleaned the heat sink blades and other bits and also changed the 3volt lithium button battery but without any difference.
Can't see anything that obviously blown or burnt out on the motherboard and all wiring has been checked and double checked so my next plan of action is to take a punt on a new power supply even though it appears working, failing that I guess it time for another computer, the current one is about 4 years old and has been worked hard.

[#10]

what about the 12 volt and 3,3 volt rails

[#11]

Also look for any electrolytic caps on the motherboard that are bulging on the top. If they are then they're duff.

[#12]

All the electrolytic caps look ok and I can't see any obvious tracking on the motherboard.
The 12volt and 3.3rails are showing as ok however on the motherboard there is a  red LED and  my understanding is that it should be lit when power is supplied to the motherboard however it is not lit even when the computer starts up for a few seconds.
There are no bleeps when it starts up for a few seconds which I understand can be helpful in trying  to diagnose the problem.
Fortunately all my important stuff is on an external drive  so there is no  loss of  important software.
Will try with a new power supply when it arrives but if that is not successful  then I,m in the market for another computer and will be seeking the advice of forum members on what they recommend.

[#13]

Reseat the video card? 

[#14]

Will give it a go.

[#15]

Try a new video card - even a cheap one....