Mild winters, such as maritime Pacific NW do not require a great deal of thought to staying warm and comfortable when running. Sometimes just the addition of a tracksuit top and bottoms is enough, but on the colder, dry days you may want to think of dressing in layers for the torso such as two wicking T-shirts, one long sleeved next to the skin and another short sleeved over the top then your tracksuit or other outer shell. Gloves become important to protect your finger tips from the numbing cold air and a hat or headband of some description to hold heat in. Remember, you lose 30% of heat through the head!! Getting too hot whilst running in winter is easy to deal with just by removal of some item(s) of clothing and either tying it around your waist or stuffing it in a pocket or sack. Feeling frozen on a run is most uncomfortable and potentially life-threatening.
Harsh winters require a little more attention to detail regarding warmth and wicking materials. More thin layers of clothing are better than a thick layer. Wicking is doubly important to carry sweat away from the skin especially if you stop for a rest in sub-zero temperatures. Ski gloves and hats can be used as they are designed to keep extremities warm in harsh conditions.
I have run in snow blizzards in freezing temperatures and my running apparel consists of wicking shorts, two wicking T-shirts (one long sleeved, one short sleeved), thin long sleeved high necked fleece, lightweight thin wind and shower-proof outer shell, wicking leggings (under the shorts), a pair of wind and shower-proof full length zipped outer shell over pants plus ski gloves and beanie hat. Even with that lot on it took over 20 minutes of exertion to feel warm and start some sweating. Footwear was normal, just the one pair of wicking socks and my trail running shoes and no problems with cold toes as my feet were under motion all the time. If I’d stopped though I think my toes would have succumbed to the draining effects of cold snow. So carrying an extra pair of dry socks might be a good idea.
Air activated hand and feet warmers can be carried with you to use as necessary. They are readily available from sports shops and hardware stores. Some ski socks have small battery powered heating elements in them to keep toes warm; that might be too extreme for running though as you build up heat due to action. The feet inside plastic ski boots are basically inactive compared to running. So, as you can see there are endless choices and something for everything.