It's warmer than sitting on a ski lift!
Salt Lake City's mild valley conditions make for reasonable bicycling on most winter days. Winter bicyclists will want to be careful not to overdress; you'll warm up as you pedal. Dress in layers so you can adjust for temperature. A thin balaclava for under your helmet in highly recommended. You will also want to have good windproof gloves and footwear.
Consider hand / foot warmers (as used by skiers) for extra-cold days. Check out this video for tips on dressing for winter bicycling from a "hard-core" bicyclist in Chicago -- a woman who dresses in ordinary street clothes appropriate for the office environment while bicycling in zero degree (Fahrenheit) temperatures.
Also, this video created by our very own Salt Lake City Bicycle Collective offers good advice for winter cycling.
Yes, there are studded tires for bikes
Pay attention to snow and black ice (clear ice that is not easy to see) on roads. Studded tires, available at local bike shops, provide good traction on icy days.
Keeping going on bad air daysIf air quality is a concern, filter masks are available as well, and may be available at local bike shops.
Further winter cycling tips available online at IceBike - Home of the Winter Cyclist and Other Crazy People.