My computer shut off without warning.  It WILL NOT turn back on.   Is this the power supply FAILURE?  I would think that if it were heat related, it would turn back on.  I've checked all the cables and outlets, and have not blown a fuse.

Puzzled.

If its completley dead (no FANS or anything running) then if i was you i would first check the power supply. Have you got another computer to test it in. If you find it isnt that i would next LOOK at the cpu Heatsink and fan.Chris:

Could be the power supply.  First, test the ELECTRICAL outlet by plugging a lamp into the socket.  

Power supplies have FUSES that are designed to blow if your system is demanding too much power or if your home experiences a power surge.  Don't replace the power supply unless you know what you are doing.  If you do replace it, be sure to unplug the computer, first.

If replacing the power supply does not fix the problem, you may have to replace the motherboard.  Check the motherboard for any signs of damage (for example, discolored or cracked chipsets).

Do you hear any sound at all or see any other signs of "life" when you press the power switch?

Was it smoking, glowing, or anything unusual?

Anything at all?