2951 TA: Aragorn learns his name and meets Arwen

In the year 2951 of the Third Age of Middle Earth, Elrond of Rivendell came to Aragorn son of Arathorn and told him of his lineage. Also in this year, Aragorn son of Arathorn met Arwen Undomiel, daughter of Elrond of Rivendell. 

   'But when Estel was only twenty years of age, it chanced that he returned to Rivendell after great deeds in the company of the sons of Elrond; and Elrond looked at him and was pleased, for he saw that he was fair and noble and was early come to manhood, though he would yet become greater in body and in mind. That day therefore Elrond called him by his true name, and told him who he was and whose son; and he delivered to him the heirlooms of his house.    "Here is the ring of Barahir," he said, :the token of our kinship from afar, and here also are the shards of Narsil. With these you may yet do great deeds; for I foretell that the span of your like shall be greater than the measure of Men, unless evil befalls you or you fail at the test. But the test will be hard and long. The Sceptre of Annuminas I withhold, for you have yet to earn it."   'The next day at the hour of sunset Aragorn walked alone in the woods, and his heart was high within him; and he sand, for he was full of hope and the world was fair. And suddenly even as he sang he saw a maiden walking on a greensward among the white stems of the birches; and he halted amazed, thinking that he had strayed into a dream, or else that he had received the gift on the Elf-minstrels, who can make the things of which they sing appear before the eyes of those that listen.   'For Aragorn had been singing a part of the Lay of Luthien which tells of the meeting of Luthien and Beren in the forest of Neldoreth. And behold! there Luthien walked before his eyes in Rivendell, clad in a mantle of silver and blue, fair as the twilight in Elven-home; her dark hair strayed in a sudden wind, and her brows were bound with gems like stars.    'For a moment Aragorn gazed in silence, but fearing that she would pass away and never be seen again, he called to her crying, Tinuviel, Tinuviel! even as Beren had done in the Elder Days long ago.   'Then the maiden turned to him and smiled, and she said: "Who are you? And why do you call me by that name?"   'And he answered: "Because I believed you to be indeed Luthien Tinuviel, of whom I was singing. But if you are not she, then you walk in her likeness."   '"So many have said," she answered gravely. "Yet her name is not mine. Though maybe my doom will not be unlike hers. But who are you?"

Excerpt taken from 
Appendix A>>Annals of the Kings and Rulers>>1. The Numenorean Kings>>(v) Here Follows Part of the Tale of Aragorn and Arwen