In a vision witnessed by Bran Stark and the three-eyed Raven, Ser Arthur Dayne wields Dawn alongside a second sword in his fight against Lord Eddard Stark and his bannermen during their encounter, slaying most of them and wounding Lord Howland Reed. He duels with and disarms Ned Stark, but before he can kill him, Howland Reed stabs him through the back of his neck. Ned Stark then takes up Dawn and slays the mortally wounded knight with his own blade.
When Bran continues his vision of the Tower of Joy some weeks later, he sees Ned holding Dawn, still covered in Ser Arthur Dayne's blood, as he climbs the stairs of the Tower of Joy. Ned then rests the sword at the foot of the bed in which Lyanna Stark is laying.
In the books
After the showdown at the Tower of Joy, Eddard returned the blade to House Dayne at Starfall; he presumably does this in the TV series as well, as he makes sure to take the sword with him when he leaves the battle.
According to George R.R. Martin, Dawn remains at Starfall castle during the War of the Five Kings, because no new warrior of House Dayne has yet proven himself worthy to wield it as a new Sword of the Morning.
In the novels, Dawn is a Greatsword similar to Ice, which Ser Arthur wields with both hands, whereas in the show, the Greatsword (which seems to be scaled down in order for the Actor to work with two swords at once) is wielded alongside a second blade in the off-hand in order to display Arthur's skill at swordsmanship. There is no mention in the novels that Ser Arthur ever used Dawn in this fashion. With no desire to narrate a lengthy exposition on Dawn and the Sword of the Morning, this change in combat style was likely made to enhance the abilities of Ser Arthur for the show, since dual wielding is a popular and powerful technique in fantasy RPGs. Dual wielding is actually quite difficult in real swordplay, unless the blades are both relatively short, or one blade is long and the other very short (in which case it is usually a specially designed parrying dagger). Again, this is assumably done in order to draw attention to Ser Arthur's prowess.