Running in the cold can certainly be dangerous to runners who do not take the proper precautions to make sure they are wearing the right clothing and have essentials easily accessible to help ensure their safety or ensure they have a way to get help if they need it. Additionally, if you know you have asthma or another condition that can make breathing difficult, running in the cold might not be the best option for you. Similarly, if you know you are especially prone to getting bronchitis or bronchitis-like symptoms from running in the cold, you should stick to cross training or indoor training during the colder winter months.
Having said all that, as long as the temperature is not nearing or below zero degrees Fahrenheit, and you are healthy, it likely will not hurt you. Make sure that you dress appropriately. The secret to dressing for cold weather running is to layer up. Layers ensure you stay warm, but can easily be shed as you start bringing your heart rate up and warming up. Winter also means shorter daylight, so plan ahead to make sure your runs outside only happen when there is adequate sunlight. Run with a friend to avoid emergencies, and no matter what, always bring your phone or a way for people to get in touch with you and vice versa. If you start feeling yourself getting too cold (i.e. numb extremities, trouble breathing and pain in your lungs, blue lips, unable to warm up even after a couple miles), immediately phone a friend to come get you and take your runs inside. Finally, be sure you stay hydrated.
Colder weather often means dryer air, as well. Dry air can wreak havoc on your skin, which can lead to itchy, broken skin and painful chafing. Fight back against some of that dryness by drinking plenty of water and a regular moisturizing routine (before and after your run). If you do not have a running belt, definitely invest in one. This will give you something to hold all of your essentials, including bottles of water to drink from during your run.