To me, snow and cold are two related, but different animals. I'll start with how I deal with cold first:
Three words: Layers, gloves, and hat.
Layers give you options so that you can run in pretty much any weather. Look for synthetics and/or merino wool as these will keep you warm even when they're wet.
Gloves will protect your hands from getting too cold. I like lightweight running gloves rather than fleece ones but I also don't run when it's bitter cold anyway.
As much as 20% of your body heat escapes through your head so a hat will keep that heat in and help you stay warmer overall.
All that being said, you still want to pay attention to the conditions when choosing to go outside and run. For obvious reasons, running in icy conditions is not a great idea nor in conditions where there's low visibility. When its really bitter cold (like sub 20 degrees F) frostbite becomes a serious problem for your nose and fingers and you have to pay attention to how they're feeling as you run.
So I don't really like running when there's snow on the ground unless I'm in a park and the trails are completely clear. Snow on the ground generally means that roads are slick and so running near a road means there's an increased chance that a car might go out of control and head towards the sidewalk. Also, snow hides a lot of things: uneven pavement, downed branches, ice, etc so the risk of tripping and falling also goes up.
Hope this helps!
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