Most manufacturers say your camera shouldn't be used in weather below 32 degrees Fahrenheit or 0 degrees Celsius, but the cold weather will have a bigger affect on the batteries than the camera. You do have to worry about taking the camera from an area of warmth to a location below freezing. When you do this, you risk the chance of having condensation form on your lens, and potentially inside of your camera body. An easy solution to this is to slowly acclimate your camera to the temperature change, but sometimes this is not always an option. When I was in Chicago for Thanksgiving this year, I tried out a new alternative, an airtight bag. When I would finish shooting at each location, I would jump in the car and promptly seal my camera in a Ziploc Bag. Occasionally when I wouldn't deem the bag necessary, I had to wipe the condensation off of my lens, but I never experienced any issues when the camera was in the bag. If the bag isn't helping, try placing a moisture wicking object in the bag with your camera such as a silica gel pack.
Once you have your camera handled, you need to make sure it stays charged throughout the shoot. The batteries do not perform nearly as well in the cold climate and a spare is almost always necessary. Keep the spare in a warm place such as the inner pocket of your coat. When your battery dies replace it with the spare and place the dead one in the warm place. Your battery is most likely not completely dead and will have a little bit more charge once it is raised to a moderate temperature.
You shouldn't prevent yourself from doing what you love this Winter, but it would be a tragedy if you ruined your camera while doing it. Keep your gear protected (and yourself) and have a wonderful photography-filled Winter!