In my experience, it generally works fine if I simply use cheap, lightweight water bottles (e.g., a 2-liter soda bottle), and put them inside my pack while I'm hiking. The surrounding material in the pack insulates the bottle from the cold air, and my body heats up the pack, so the water doesn't freeze. If the weather is very cold, I can use extra care in packing my pack so that the bottle is close to my body.
Some of the answers have suggested putting the bottle inside one's jacket or clothes. This seems like it would be extremely uncomfortable and impractical, especially with a large water bottle. In my experience (down to about 0 F or -20 C), this isn't necessary, and the water stays unfrozen inside the pack, if it's close to my body.
If I need water to stay unfrozen overnight in cold temps, I can put the bottle inside the tent or even inside the sleeping bag -- but in this situation I'm probably depending on melted snow for my drinking water anyway.
The most common problem I've seen during the day while hiking is that when people use platypus-style bladders with a mouth tube, the water inside the tube freezes. This is one of the many disadvantages of these systems, and it's one of the many reasons I don't use them.