What is your essential winter training gear?

I just placed an order with DeSoto the other day.

I have in there some fleece arm warmers, a hoddie kinda-thingy and some other stuff.

Once I get my items I will write some reviews on BT. 

Though keep in mind winter in Texas is not the same as winter in Alaska ;-)

A gym membership and a bike trainer! 

If I'm going to be crazy and brave the Chicago winter, then a LOT of pieces that can layer.  Long-sleeve tech gear, long running tights, long bike pants, booties for over my bike shoes, balaclava, skull caps, gloves that have the attachment to make them mittens to keep feeling in my fingers, thermal socks, wind-breaker jacket and the little hand warmers that you can put into your gloves if needed. 

And hot chocolate for when I get back inside.   

Is there a difference between running arm warmers and bike arm warmers?  I have a trainer and will ride indoors all winter long until spring. I ride exclusively early in the warming while its dark, so I'll be trainer bound.  I have trainer road and some sufferfest videos.  I'm good there, but I will be running outside.  

A neck warmer goes a long way on windier winter days.  I like double layer socks for running - they are thicker and also don't get too soaked and cold when accidentally stepping in a slush puddle

Fleece lined tights are quite nice as well.  

Booties for the bike if you're going to even do some later fall rides

I don't know, but highly doubt there are any big differences.

My all time favorite winter item is my North Face Apex Bionic jacket. It's a very light stretchy jacket with lots of reflective piping. It's fitted and is wind blocking but still breathable. It also has the thumb loops which help keep my hands warm.  Great for a run or ride in cool weather. Also great as a layer under a heavier jacket on super cold days.  It was expensive, but well worth it to me.  I don't wear it on my mtb as I would cry if I wrecked and tore it.  I don't know if trisports carries them, but figured it was worth a mention.

http://www.backcountry.com/the-north-face-apex-climateblock-jacket-womens

If you've got a code for TriSports.com and want some nice winter clothing look at their Pearl Izumi page, especially this:

http://www.trisports.com/pearl-izumi-mens-pro-thermal-ls-jersey.html

and;

http://www.trisports.com/pearl-izumi-mens-elite-softshell-jacket.html

Those are both super nice. The later has impressive cross over characteristics- it works for bike and run and even casual wear. A lot of value.

 I have lots of winter stuff, what I use most  Is my NB running Vest, I prefer it over a full Jacket but I do have a jacket as well, a balaclava for sure & touque, Goretex waterproof trail runners these are pretty much indispencible around here keep the feet dry and offer better traction in light snow over straight runners , drymax  or mirano wool socks, good  light warm gloves, then its layers from there, My Cwx Insulator Tights are great when it gets really  really colds too. I got a couple of the Asics shirts already mentioned  last season too they are pretty warm and seem to be holding up well.

SO lots of options to spend your cash on

For winter running:

WOOL SOCKS are essentials.  I can run through a slush puddle and not feel any cold or wet after the first second.  I prefer thicker ones and they aren't anything special.

A gaitor

I do prefer my fancier running hats, like one from Smartwool and another from Pearl Izumi made out of synthetic material. I like a light layer on everything.  They keep my ears warm and wick the sweat.

Trail running shoes.  I wear them when the sidewalks/streets are not cleared for better traction and they keep out the cold/wet a lot better than my regular shoes.

Strap-on traction helpers for my shoes.  I use these exclusively for icy days,  otherwise the trail shoes do the trick. I.e. Yaktrax.

The rest I make up with what's clean for the day, layering different shirts, jackets, and plain old cotton stretch gloves from Target.   I've had several different brands of running tights (LOVE tights).

I could make a much longer list for winter biking...

Arm warmers, gloves and a hat for when it's raining or a headband that covers the ears when it's cold. Also, a good pair of tights.

Extra Bonus: wool running socks if your feet are going to get wet

I also don't know if there are different running and cycling arm warmers, but I have two different pairs, one just standard material and the other is fleece lined.  I will wear both for cycling depending how cold it is, but mainly just will wear the standard pair for running.  I think I get warmer running so haven't needed the fleece for running.  I prefer to run or ride in a vest plus arm warmers rather than a jacket, gives me more options.  But, I live in Northern California, so our winters aren't as cold as other places.   I guess if it were colder, I'd end up training in a jacket.

Wanted to add:

Headlamp and Blinkie lights for beeing SEEN in the dark.

Beanie cap, "warm gear" long sleeves, windbreaker, gloves. I started wearing running tights last winter after much resistance and was glad for it, I will throw a pair of shorts over top.  If really cold (for me 20s and below), I will just throw on whatever I have to keep me warm, hoodies, a few shirts, etc...  I got a pair of arm warmers for christmas, but I really didn't like them too much, perhaps they were too tight.  I do like the idea of being able to pull them down if you warm up too much.

I have cheap gloves, cotton type, that I can layer.  They are great for "snot wiping" when I wear windbreaker type jackets have a tendency to smear more than wipe. 

trainer and sunglasses for running. winter sun is pretty bright.

luckily i can wear shorts/t shirt for pretty much all my runs through the winter.