The body generates waste heat during heavy exertion, so running in the winter cold is actually less difficult or dangerous than running in hot weather. With proper winter running clothing and some common sense it is possible to continue running through the worst winter weather.
As for any outside activity, the key to running comfortably in cold weather is to dress properly. Running in cold weather however does not require as much clothing as more sedentary cold weather activities. A good rule of thumb is the 20 degree rule. When running, dress for weather that is about 20 degrees warmer than the actual temperature. This rule is not exact, but is a good guideline. Wet or windy conditions require dressing more warmly than the rule would suggest. With experience individual runners will learn how much clothing they need for different temperatures and other conditions.
To keep warm enough, but not too warm, in any temperature dress in layers. It is then relatively easy to add or remove layers as needed during a run. To avoid feeling wet, clammy, and cold the innermost layer should be fabric that wicks moisture away from the skin. Many modern technical fabrics accomplish this without the itching that comes from wool, the best natural fabric for this purpose.
An outermost layer of Gore-Tex, or similar fabric, will block wind and rain without sacrificing breathability. Running clothing made of technical fabrics works very well, but can be expensive. Runners on a budget who are willing to sacrifice a little comfort can also get by with old fashioned sweat suits.
Wear a hat. A stocking type hat provides extra warmth when needed yet easily stashes into a pocket when not. Gloves will keep hands warm, but pulling long sleeves down over the hands also works. Because runners generate heat in their legs they need fewer layers on their legs than on the less active parts of their bodies.
Wind often exacerbates cold winter weather. When dressing for winter runs take the wind chill as well as the temperature into account. Winter running clothing and running pants should include wind proof outer layers. Plan running routes to avoid running for long periods into the wind.
Special risks of winter running include snow, ice, and darkness. When running on snow and ice, be careful not to slip and fall. Forgo speed training and run a little slower to reduce the risk of slipping. Also be aware of traffic. Drivers will also be slipping, so pay attention and keep a safe larger than normal distance from traffic.
During severe blizzard conditions, it is wise to stay close to home. That allows ending a run early if necessary.
Running in deep snow is more difficult than normal. It also provides a harder workout. Modify your running schedule and speed expectations accordingly. On days of deep fresh snow, consider cross country skiing as a substitute for running. It provides similar aerobic benefits and a better upper body workout.
Shorter winter days force many runners to run in the early morning or evening darkness. When running in the dark, take extra safety precautions to see and be seen.
With proper winter running clothing and an occasional weather dictated modification in your running schedule, you can continue to enjoy running outdoors all winter.