Is it normal for the cooling fans to only run intermittently when the AC is turned on, or should they run continuously?

Also, should the the cooling fan continue to run after the car is turned off, when the AC has been running?

I replaced the fan relay 6 months ago so I want to be sure it's still working correctly. I replaced it because the coolant fan kicked on as soon as the car was started (hot or cold) and continued to run at high speed after the engine was off. (during the winter when AC was NOT on)