Can reindeer really fly?
Generally, reindeer can't fly. Reindeer are mammals, and the only mammals that can truly fly are bats. There are some species of squirrels and possums which have membranes of skin between their forelegs and ankles which they use to glide, but they are not in control of their 'fall'.
Except - Santa's reindeer are different. It is said that the reindeer that pull Santa's sleigh on Christmas Eve can fly.
Whether it is only on that one night, once a year, or whether they are special reindeer which can fly all year 'round (not that they need to) is yet to be determined.
See also the related question.
Yes, reindeer are real animals which naturally live in the Arctic, although they can now be found in many areas of the world. They cannot fly but they can be trained to pull a sleigh across the ground. In the US, reindeer are known as caribou. You can see more information about reindeer by clicking on the link in the Related links and sources section, below.