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.