I've got a home assembled PC about 18 months old. Asus motherboard with an
Athlon dual core processor.
For the last few months it's been shutting down at random. Thinking it
was the PS replaced it. No improvement.

To help eliminate whether it's a hardware or software issue (almost
certain this will be hardware related though), you could try booting
up a live linux CD and seeing if it lasts any better. If not, it's got
to be hardware related...

I suppose there are a couple of random things it might be, such as
faulty RAM* (google MemTest86), or a known fault in the BIOS causing
timing issues or somesuch, but I doubt it, given that you said it
improved for a while.

* my worst weekend ever consisted of 26 failed installs of XP, each
failing at a random different point, with a random different error.

Any other guesses?

About the best I can come up with would be a dry joint on the board.

If it was an older machine i'd suggest the capacitors, but yours is
probably from the time they first went over to RoHS soldering methods
- which I believe are prone to this sort of crap :-}