Bilbo's Mithril Shirt was a shirt of chain mail, alternately described as a vest and a coat, made from mithril. It was moved several times during the course of the series, from Smaug's hoard to Bilbo to the Mathom-house to Frodo. It was taken by the Orcs of Cirith Ungol from Frodo, but was later with the Mouth of Sauron, as he showed it to Gandalf, Aragorn, and the other leaders outside the Morannon in the Battle of the Black Gate. It was later returned to Frodo.
The origins of the shirt are largely unknown, though it was said to have been made for a young Elf prince long ago. It eventually came to be part of the Dwarven treasure hoard of the Lonely Mountain. It was found by the dragon Smaug and added to his treasure heap. Here, it was given by Thorin II Oakenshield to the Hobbit Bilbo Baggins, who wore it during the Battle of Five Armies. After returning to the Shire, Bilbo loaned the shirt to the Mathom-house. Before departing the Shire for Rivendell, Bilbo reclaimed the shirt and wore it on that journey.
Many years later it was given by Bilbo to his nephew Frodo Baggins as Frodo set out on his quest to Mount Doom to destroy the One Ring. Frodo wore the chain shirt under his tunic unknown to the rest of the Fellowship. It was discovered when, during a skirmish in the mines of Moria, an Orc chieftain stabbed Frodo with a spear (Alternatively, in the Peter Jackson film it was a troll who speared Frodo). The shirt prevented what would have been a lethal wound, leaving Frodo badly bruised but otherwise uninjured.
The shirt was confiscated from Frodo in the Tower of Cirith Ungol. The two squadrons of orcs that captured Frodo fought over the valuable shirt, and most were killed during the conflict. One of the two Orc leaders, Shagrat, survived the fight and seized the shirt, escaping the tower with it. The shirt eventually ended up in the possession of the Mouth of Sauron at Barad-dûr, where he presented it to Gandalf at the Black Gates as false evidence that Frodo had failed in his quest. In his wrath Gandalf retrieved the shirt. The wizard later returned it to Frodo after the destruction of the Ring.
Following Frodo's departure to the Undying Lands it is unknown whether he took it with him or left it to Sam like he did with the Red Book, in which case it likely would be in the possesion of the Fairbairns.