Why do horses not sleep?

Horses DO sleep!

They just mostly do it standing up!

If you walk up to a horse before sunrise you should make sure to make a bit of noise while approaching, lest you startle it and wake it up on the wrong side of the paddock! (Without even noticing it was asleep!)

Some horses will lie down for a nap... especially in the sun on a cool day, or the the hay of a stable on a cold night!