This is valid for jet aircraft:
First, there's survivability at low temperatures... that's when an aircraft , de-energized, spends the night in a very cold place (not operating, just surviving), for example. Fuel and water reserves will probably have to be drained after landing. Then there are limits to what temperature components will endure in order to be able to start when required. If the limits are exceeded, additional proceedings will be necessary when starting up. If the temperature is within the aircraft's operational envelope, usually that's around -40º C on ground, you may re-fuel the aircraft and fly. If it's not, you don't fly.
If it's really cold (maybe around -50º C), operation is possible with a few tricks to keep components warmer... APU remains ON
and the aircraft energized, for example... That will keep all the equipment inside the pressurized region in adequate conditions. Hydraulic pumps may require to be turned on and off from time to time... Engines too, in order to keep the fuel from getting too cold. Doors or access panels may need some hot air blowed.
In flight, operation might be extended to -55, -60 or even -70º C... In these conditions, the components are not always at the same temperature as external air due to Mach effect.. that's heating caused by friction. So there might be a note in the flight manual restricting operation to a determined Mach range if temperature is too low.
Lots of problems and solutions involved... and not only engines