Getting excessively cold or wet is not fun. In the winter it is often something that puts runners off going out and getting their training done.
However, this shouldn't be an issue as there are waterproofs and winter kit available on the market which will be good enough to keep you at the right temperature even on the coldest of days.
Goretex is the most famous of the breathable materials that these jackets and trousers are made off. Whilst it is quite expensive, it does do a great job at keeping you dry, both from the rain and also, with it's breathable qualities, for the perspiration that you produce whilst training.
Other materials are also used, and the key is to look for kit that is lightweight, has venting systems (even if the material itself isn't breathable) and is at least shower and windproof even if it isn't fully waterproof. As with other external items of wear, if you are going to be running at night, having reflective strips on your kit can be useful.