When it gets extremely cold, even being outside can be dangerous. If you are going to run, drink plenty of water, carry your cell phone, be careful, and please, dress accordingly.
1. Head. We lose around 10% of our heat from our head. The solution? Wear a hat or beanie. If it's below freezing, consider wearing a balaclava or face mask to cover your head, neck, ears, basically everything but your eyes.
Tip: If it's cold and windy, break out the Vaseline. You can put it on your lips, nose, cheek to prevent chapping & windburn.
2. Upper body. Layers are the only way to go. Body heat gets trapped between layers, warming you better. But, if you do start to get warm you can easily shed outside layers. Your base layer should have moisture wicking properties -- you don't want to start sweating and stay wet. Go for a synthetic or special wicking material; do not wear cotton on your bottom layer, it will not dry. Your second layer should be insulating. It should also wick away moisture but you want it warm enough to trap your body heat and help keep you warm. Fleece is a good option. Finally, your third layer should be wind and/or waterproof, like a Gore-Tex material.
3. Hands. It's vital to keep your hand covered, as much as 30% of your body heat can escape through your extremities. You want moisture wicking running gloves or mittens.
4. Lower body. Synthetic running tights or pants are perfect. They will wick away moisture and keep you warm. If it's especially windy, throw a windproof pair of track pants on top of them.
5. Feet. Stay away from the mesh. Those shoes that are wonderfully breathable in the summer can be downright dangerous in the winter. Water and cold will come in through that mesh. Personally, I like waterproof trail running shoes for any snow, slush or rain running. They provide more traction and keep my feet dry. Don't wear cotton socks. I feel like a broken record, but you want moisture wicking. Try acrylic or even wool.