Many are designed for extreme temperatures now, but they are not efficient in such temperatures due to the properties of the refrigerants available. It will usually still run, but will run longer than usual and may not be able to keep up with heat demand. During a cold snap, might run continuously and may need frequent defrost cycles which further reduces efficiency and heating capacity. How cold they can/should go depends on things like sizing, refrigerant, building insulation, solar heat gain, and other factors. In a country like Canada, air source heat pumps will always be installed with a backup heat source. In Canada, that is most commonly the same natural gas furnace that the house was always originally equipped with, while the heat pump replaces the air conditioning unit, but backup heat can also be resistive electric, heating oil, wood, and more, and a decent thermostat will have an outdoor air sensor that can detect an appropriate outdoor conditions at which to switch from heat pump to backup/"emergency" heat when needed, and then back to heat pump when conditions permit.

In my experience, my heat pump provides about 90% of my heating from fall to spring. Only a few severe cold snaps put the heat pump on the bench for a few days switching over to emergency heat completely, and most of the winter it runs frequently to continuously to maintain consistent temperatures and only needs a short and occasional boost from the furnace. The gas backup is also nice peace of mind in case of a prolonged power outage due to a potential ice storm, as the furnace blower and electronic controls can easily be run off a small battery pack or generator for a very long time.