Radiator fan stays on all the time and car runs...

The fan relay may be malfunctioning (the fan relay is what actually switches the fan on and off and gets its signal from the ECM, which in turn gets its signal from the temp sensor) testing/replacing the relay and temp sensor would be the best thing to start with,... unless you notice any additional (non factory) wire attached to the power wires to the fan.  Sometimes folks, in an attmpt to fix  an overheating or fan relay problem, attach a ignition switched positive lead from the fusebox or other switched source to the positive poer lead on the fan,.. making the fan run whenever the ignition is switched on.   

There is a thremistor that tells when to turn the fan on when the temp reaches a set temp, usually over 200 degrees F. That could be the problem. It is a wire connection that goes to the engine or radiator.

Also the AC is in control of the fan. When you turn the AC on the fan turns on. Might be something going on there.

Asks parts guy at your local store

Try searching for a wire that goes from the fan to a another electrical sourse that is only live when the vehicle is running. Like the coil, distributor. Goodluck and PEACE