Riding your bicycle during cold weather conditions or Winter is a subject that divides the opinions even here in the Northern Europe. Some like it and some don't. Some think it is possible, some think the ones who ride during the Winter are totally out of their minds.
In my humble opinion, it is not so big a deal at all. There are few things you have to be aware of and some precautions to be taken.
1. The Man
Lets start with the subject of protecting oneself against the conditions i.e. dressing up.
There are three basic rules to be obeyed: Layers, Layers and -- guess what? -- yes, Layers!
Protect your extremes - feet, hands and head. Wear layers, the under layers that can take the humidity away from your skin, wool or some synthetic material to transfer the humidity outwards and on the outside layer wear something that keeps the wind outside but let's the humidity out.
Cotton is generally considered not good at all since it just gets wet and that's it.
I usually wear long legged underpants and a pair of jeans on my daily commuting of 7,5km (4,7miles) to the office and then another 7,5km back home. If the weather is extremely wet I put another layer of Drymax-pants on top of the jeans. This keeps the wind/wet out nicely. Two layers of natural fibre (i.e. wool) socks will do for the feet. (plus leather shoes) Two layers of shirts (usual short sleeved as the first layer) on top and then a winter jacket. Scarf to protect the neck/throat and a buff to protect the cheeks. Ear flaps for the ears, and a helmet of course. One could use an army style helmet commando hat also. On my hands I have long winter gloves and then warm leather "mittens" on top of them. If you ride with SPD pedals, use winter shoes or protect your summer shoes with neopren covers.
2. The Machine
Usually people wish to have slightly different ride/bike on Winter for the matter of there might be snow and ice on the road.
Thus the winter tires, either knobbed or studded tires will do. I have studs in both ends, some have them only in front (it is more important to not lose grip in front - if you lose it, you will fall very easily). Mine are 35mm wide on 700c rims - in front I have a Nokian Hakkapeliitta W106 and in rear a 35mm Schwalbe Marathon Winter. Both are ok, but Nokian takes less pressure and thus makes it a bit "slower" to ride.
And since the speed is slower, at least the ones riding single speed bikes tend to select slightly shorter gear too. Then - of course depending on what kind of winter we are talking about - is the matter of mudguards/flaps. If it is dry and cold - you don't really need them. If it is close to zero and there are times when the snow melts away - then it is not a bad idea to have them mounted on a bike.
If you ride with gears or use brakes, then make sure to keep the cables lubricated since if the water / humidity freezes inside the cable hose, then the brake/gears become unusable.
On the bearings one should have such a lubricant/grease/vaseline that can handle the lack of heat (cold weather) without getting tacky or slow. I remember years back when I was still living in a town in Eastern Finland called Joensuu one Winter, the temperature was -35'C in the morning and as I pedalled to the office down town I noticed that the Shimano LX hubs were not rotating very well at all - at least the man warmed up a bit more ;)
EDIT: one more thing - See and be seen! Use lights, use reflectors. If you wear dark and ride in a dark - the car drivers and other users of the road will not be able to see you. You can get pretty effective led lights if you need to see more, and they help the others to spot you too!
3. The Attitude
It is possible. It can be fun. It is generally pretty safe. But, it takes longer than during the Summer. You have to foresee the possible hazards in advance. You have to observe the other users of the roads more carefully. It can be slippery, it can be wet. Remember that if there is wind, the weather feels much colder than the numbers show. I have been riding all year all of my life, first in inlands town Joensuu and then in a city by the sea, Helsinki.
But in general - it is fun and it is possible and if you feel like it, you should give it a try.