In the fictional world of Middle-earth created by J. R. R. Tolkien, Rhûn is a large region of eastern Middle-earth. Rhûn (which also the Elvish word for "east") is the name used for all lands lying east Rhovanion, around and beyond the inland Sea of Rhûn, whence came many attacks on Gondor and its allies during the Third Age of Middle-earth.
Almost nothing can be known of the lands beyond the Sea of Rhûn from Tolkien's written work, though it is vaguely described in The Fellowship of the Ring as a realm of "wide uncharted lands, nameless plains, and forests unexplored". The wizard Gandalf had never explored there, and though Aragorn is said to have travelled there, there is no report of his doings.
Rhûn's ancient geography can be gleaned a little from The Silmarillion; throughout most of the First Age the vast Sea of Helcar was located here and beyond that the Orocarni ('red mountains'). Somewhere in the east, too, lay Cuiviénen and Hildórien, where Elves and Men first awoke: all the Children of Ilúvatar could trace their ancestries back to the eastward regions of Middle-earth.
Rhûn was the domain of the Easterlings, Men of Darkness who were ready to follow both the Dark Lords and fought as their allies in war. These lands, too, were peopled by lost Elves, Avari and Úmanyar, and by four of the seven clans of the Dwarves.
During the Third Age, Rhûn was visited by three Wizards; Saruman, Alatar and Pallando, and though Saruman returned into the west, the two Blue Wizards remained or went to the south to the lands of Khand and beyond. Sauron himself journeyed into the eastward lands, hiding there from the White Council during the centuries known in the west as the Watchful Peace. He seems to have had a second fortress somewhere in Rhûn where he dwelt during that time and earlier during the Second Age, as his original realms were in the East, not in Mordor. This eastern abode of the Dark Lord is briefly noted in The Silmarillion as a place which he surrounded with fire.