They are said to have arrived not in the Third Age, but in the Second, around the year 1600, the time of the forging of the One Ring. Their mission was to weaken the forces of Sauron and it is said that the Wizards had a pivotal role in the victories of the West at the end of both the Second and the Third Ages.
It is however possible that they ultimately abandoned their roles and started magical cults.
In Letter 211, Tolkien says that the two wizards went into the East, and likely failed their mission, perhaps having started magical cults; however he stated that as a hypothesis, not a fact. Later in his life he elaborated more on their characters.
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