2000 BMW 323 fan keeps running under hoo...

Most likely, it is the temperature sensor that activates the radiator fan - (If it stays on all the time) or less likely; a faulty relay - however, I'd bet that there's a coolant temperature sensor that is in the cylinder head, radiator, or waterneck - responsible for cooling fan operation . There is more than one temperature sensor, so be sure you replace the correct one - The one that activates the radiator (cooling) fan relay - One of the others is for the ECM (computer) to deternine injector pulse width (Drivability) and cold/hot running/starting) and yet another for the temperature guage or light inside on the dash - Thus - the cooling fan temerature sensor is what you need to look at, and the system is wired to allow the fan to stay on after you turn off the car so it can cool down...(there is a testing method - using a meter if you are familiar) but they commonly fail and cause the same problem you are having...Your battery is in good shape, I assume, and otherwise the engine temperature is normal? If so, the sensor is probably to blame...

Hi,
First - disconnect fan and charge your battery - or you could do permanent damage to it.

With the engine cool, locate the Fan thermal switch. It should be on the radiator itself. Pull off the connection (this should turn off the fan)

Check the two terminals with a meter as they should not conduct when cool.
Fit a replacement. You may loosa a little coolant - but not too much if you are quick.
Top up with similar type.

If the fan switch is ok, it may be a relay which has stuck. This should be located at the end of the fan wiring loom.