The following a some of the general mythologies & stories, not related to specific species. Where there are any stories related to a specific species, these will be found on the page for that species. As with other confusions between hawks & falcons, sometimes the myths about falcons can also be related to hawks.
In Ancient Egyptian mythology, Horus, the God of Lower Egypt appeared in the form of a falcon (or hawk), whose right eye was the sun & left eye was the moon. It was believed that the reigning king was the earthly incarnation of Horus. The kings were often depicted as falcon headed. The Sun God, Ra, was also depicted as being falcon headed.
In Norse mythology, Freyja, the goddess of love, fertility & sexuality and the guardian of the dead, is associated with falcons. She used a falcon feathered cloak to travel between the mythological worlds, taking the form of a falcon.
A falcon, called Vedfolnir, sat on the beak, between the eyes, of the wise, all-seeing eagle on the topmost branch of the "world tree", Yggdrasil. The falcon reported on all that it saw in the heavens, on earth & below, to Odin.
In 1405, the Isle of Man was given to Sir John Stanley by Henry IV on condition that he paid homage to him & gave him two falcons & similarly to every future King of England on their Coronation Day. This tradition continued for over 400 years, until the coronation of George IV in 1822. A falcon appears as one of the supporters of the shield on the Isle of Man coat of arms.
A Cast of Falcons