In Generations III and IV, when the player's Pokémon with Run Away faints in a wild battle, if the player attempts to flee instead of sending out another Pokémon, they will escape without fail. Fron Generation V onward, escape is no longer guaranteed in this situation.
Until Generation V, when activated, the message "<Pokémon name> escaped using Run Away!" will appear. From Generation V onward, a banner with Run Away shows up instead.
When its HP gets low, the Pokémon is Terrified and cannot get the Stuck status condition.* When its HP gets low, the Pokémon gets the Terrified status condition, but it's not affected by the Stuck status condition.*