Thinking about Stuarts thread about clothing, and thought I'd add something about other kit for the same conditions.
Cutting tools I carry a knife, a saw and an axe (full size) if I can. The saw can be the famous lapplander or a buck saw. The lapplander is good for smaller firewood and such, but no good for large firewood needs; that is the territory of the buck saw and axe. Axes are tricky in deep snow, so be carefull. I carry the knife in a thong around my neck, since the layers outside my belt can make a belt knife inaccessible.
Cooking A largish pot (2-4 L is good) for melting snow is almost essential; life is much harder without this. If you have some water to add to the pot before you add the snow things will go much faster (the water absorbs the heat from the pot much better than snow, you can ruin a pot on a hot fire before the snow starts melting. If there is a liquid portion things are much better. A lid keeps the water from getting a smokey flavour that bothers some people (and is more efficient). A large cup and spoon, but no metal that can freeze to your lips (the plastic 0.4 L kuksa is ideal in winter but some people like plastic commuter thermos cups better). I mean it about the metal; you will look very stupid and end up with damaged lips.
Sleeping I like a closed cell foam mat and a reindeer hide, but two foam mats can do as well. I've seen Thermarests refuse to self-inflate (the foam core was not very springy in the cold), and if you blow into them they get icy and non-insulating. If possible I put a layer of boughs underneath this; even some birch or willow will be better than nothing. A good sleeping bag, one that can actually keep you warm (I love my Carinthia Survival One, but for years I carried two 3-season bags inside each other; it was as warm, but a bit more complex getting the hoods lined up). A cotton or poly-cotton bivy to protect it in snow-shelters and around fires and sharp twigs: mine is a simple fabric condom, no zippers to make life complex. If you spread the bag out in good weather at least some of the frost that accumulates will sublime (i.e. freeze-dry).
Unless I'm in a heated shelter I'll bring my boot liners into the sleeping bag; this makes getting dressed in the morning much more pleasant. Generally I sleep in my wool underwear and one set of socks, with a wool balaclava on my head if it is very cold. I might bring a wool sweather into the bag as well (the bag is fairly roomy), and use it as extra insulation under my thorso if I think it is needed.
I actually keep one of the chemical hand-warmers in the pocket inside the sleeping bag; it allows me to add some heat in the foot section if I'm very cold for some reason.
Odds and ends
A head-lamp is very useful (currently it gets dark by 3 PM and sunrise is after 9 AM where I live). If possible get one with a battery pack that can be kept inside the clothes, and that can be operated with mittens (I use an old Petzl Actic). Make sure to keep it inside your sleeping bag at night.
A brush for getting rid of snow is good to have. I got a cheapo dish-washing brush and cut down the handle a bit, drilled and hole and added a lanyard; it lives in a pocket on my shell layer. Or make one from some spruce boughts and a bit of string.
Hand-warmers are usefull: I carry the coal stick based ones, two of them tucked into the front pockets of the pants will help keep your feet warm, or dry a pair of boot liners overnight (unless they get filled with snow, in which case you are worse off than before; they are now soaked instead of damp).
A shovel is essential. I carry the old Rottefella, but many others are good. Non-metal handle is good (keeps your hands from freezing), and a T-grip is better than a D when wearing thick mittens. If you loose your shovel you can wrap some fabric around a Y-shaped branch, add some water (possibly pre-drunk to save melting more snow than nessesary), and allow to freeze. It is crap as shovels go, but better than nothing.
A rain-poncho, basha or fjellduken is good to have; it does not need to be waterproof, just keep snow off your sleeping bag at night. No pegs; in snow you tie to a stick and bury it in the snow. Practice tying knots wearing mittens...
Some cordage and rope as well as repair kit comes in handy. Does your ski bindnings need a screwdriver? How could you repair a broken ski? Apecial ski-tips are available, and some thin Al sheet stock and some furniture tacks can also come in handy in such cases. I carry two lenghts of lampwick in case I need to make a pair of Roycraft snowshoes in order to get home (if you make them narrower and longer than customary they get a bit more like the Siberian snowshoe-skis, which is an improvement).
Pack or pulk If possible I prefer a pulk since I have better balance and feel less "loaded down" when using one compared to a backpack. But steep hills with a pulk can be special indeed, as can dense brush (add pulk, solid harness and skis; it can be like driving with a long trailer in an inner city). I use a pulk that is long enough to sleep in (extra insulation, mine has a precut foam pad in the bottom), or use as a roof in a trench shelter. If you have a pack make sure it works with thick clothes and can be opened/closed/adjusted with mittens. I always have a water-proof liner inside my backpack (when not using a pulk): imagine going though the ice, getting soaked, and then not having any dry clothes or sleeping bag to get into (the same applies to the duffels in the pulk, obviously).
Zippers have large loops or monkey-fists to enable manouvering with mittens. Actually, make sure that all your kit can be manouvered with mittens; you will be very frustated if you have to take your mittens off at -40 C just to get to your parka or foam pad.
Then you end up with things like map and compass (the mapcases are actually not bad under these conditions), toothbrush, camera, etc.
Bushcrafter (boy, I've got a lot to say!)
Ace post. I had a great time in the Arctic for 10 days of camping and dog sledding. Got my routine and kit quite sorted in a few days and found it weird flying back to Britain where the temps were much warmer but I reverted back to modern street clothing and was colder than in Norway at -30. Amazing what woolen underware and a couple of other wool layers can do in the cold.