Your hands are warm while climbing if you wear warm clothing. The first things the body lets cool down, and which are therefore not supplied with enough blood, are the extremities. That's a normal preservation process; frostbite usually starts at the toes and fingers. To avoid this, I try to support my body everywhere. A down jacket is an absolute must, plus Merino clothing (long sleeve and T-shirt) and long underwear (ideally also Merino). The whole thing is rounded off with down trousers that you can wear while belaying (whether short or long). Make sure here that the main tie-in loop can stick out at the front. On top of that, a warm jumper, a neck gaiter, a thick warm hat. Gloves and warm socks plus knitted/Merino socks complete the package. Good shoes with a fleece lining help to warm the tips of your toes, for example. Anecdote: I have a climbing partner who has heatable insoles that can be controlled by remote control (battery-powered, rechargeable). 

Gadgets: some clothing manufacturers offer so-called pulse warmers; these can be pulled over the hands and warm the wrists, thereby protecting the blood circulating from the depths of the body. 

Women: protect your kidneys particularly with a kidney warmer, or a scarf you wrap around, or a down skirt you can pull up a bit higher. Experience shows that this in particular allows a lot of warmth to be stored in female climbers.

Preparation: dress warmly, put on your harness and a down jacket over it, put on the down trousers. Tuck your climbing shoes into your jacket, then they're warm when you put them on and the rubber is softer. 

Most important rule when climbing in the cold: avoid injury by warming up your climbing. Find an easy route that you can climb even with cold fingers. Do lots of easy moves. Don't overdo it, but use your fingers – only then does your body send the right signals that they need a good supply. 

As a rule, the fingers get warmer after the first route; after the second they then stay warm for a while, and on the third route you can already climb with warm fingers. Beforehand, just pump blood into your fingers again with arm circles and star jumps. Then you can tackle the project too. And don't let it get you down if everything isn't immediately full of feeling and you don't feel every little bump as you do in summer. Your fingers first have to get used to it. 

Note: the less skin contact on the rock, the less warmth you give off. Slopers and jugs draw more warmth from you than edges or pinches.

Warm food seems hard to bring along, but there are great ways to warm yourself from the inside too. Some people lug a gas stove with pasta and water up the mountain. Personally, I prefer a small pack size and little weight, so I prepare a thermal container at home. Make up instant pot meals at home and take them along; they usually then stay warm for 2–4 hours. Hot tea or soup can be brought along brilliantly in a thermos flask. Likewise coffee or similar hot drinks. Bear in mind, though, that milk in thermos flasks quickly starts to turn and goes off, and then you'll have lugged it along for nothing. Look for energy-rich food; the body needs a great deal of energy to heat itself, and the more it has in reserve, the better it can work. That way it also keeps your hands warm while climbing.

If there's snow lying around, then grab a handful and rub it into water in your hands. The body very quickly notices that it's unusually cold at the hands and opens up the blood supply to the fingertips to counteract it.