The two ancestral valyrian swords of House Targaryen are Blackfyre and Dark Sister.

Fun Fact: Blackfyre and Dark Sister have yet to be mentioned in the main series.

Before they even had names, GRRM was already playing coy with their fate:

I suppose that there must have been a Targaryen Valyrian sword ..what happened to it? it is never mentioned. Was it melted down like Ice?

No Comment -SSM, The Targaryen Sword: 22 September 2000

Fate of Dark Sister

Dark Sister (made for a woman's hand) was last known to be held by Bloodraven.

GRRM confirmed that Bloodraven took the sword to the Wall with him: GRRM Confirmation

Which is odd in itself seeing that the Iron Throne no longer had Blackfyre at this point, so unless Bloodraven stole it, or bartered Egg for it, the possibility of Egg/Bloodraven working together exists.

Fate of Blackfyre

Blackfyre's fate is a little more clouded in mystery. While Blackfyre was traditionally passed down from king to king until Aegon IV gave the sword to his bastard son: Daemon Blackfyre aka The King Who bore the Sword, Dark Sister has been held by numerous characters (Maegor I and Jaehaerys I both held both swords at one point).

Blackfyre passed from Daemon to his half-brother Aegor (Bittersteel) Rivers, who refused to give the sword to Daemon II:

Frey and I harbored doubts about Lord Peake's pretender since the beginning. He does not bear the sword! If he were his father's son, Bittersteel would have armed him with Blackfyre. -The Mystery Knight

After that numerous things could have happened to it. Bittersteel could have armed any of Daemon's other sons or grandsons with it, but we do not have that info yet.

This is Bittersteel's death (unknown if the sword is Blackfyre):

Bittersteel eluded capture and escaped once again, only to emerge a few years later in the Disputed Lands, fighting with his sellswords in a meaningful skirmish between Tyrosh and Myr. Ser Aegor Rivers was sixtynine years of age when he fell, and it is said he died as he had lived, with a sword in his hand and defiance upon his lips. Yet his legacy would live on in the Golden Company and the Blackfyre line he had served and protected.

Bloodraven vs. Bittersteel

Their hatred of each other grew for so many reasons (Shiera Seastar, Bracken/Blackwood, Bloodraven's status at court due to his mother's reputation). And we know they fought two duels.

First Blackfyre Rebellion (The Redgrass Field)

This is the first (and only confirmed) time that Dark Sister and Blackfyre clash:

Daemon and his eldest sons, Aegon and Aemon, were brought down beneath the withering fall of arrows sent by Brynden Rivers and his private guards, the Raven's Teeth. This was followed by Bittersteel's mad charge, with Blackfyre in his hand, as he attempted to rally Daemon's forces. Meeting with Bloodraven in the midst of the charge, a mighty duel ensued, which left Bloodraven blinded in one eye and sent Bittersteel fleeing. -TWOIAF, The Targaryen Kings: Daeron II

Third Blackfyre Rebellion

Bloodraven and Bittersteel fight a second duel but it is unconfirmed if Blackfyre is involved:

In 219 AC, Haegon Blackfyre and Bittersteel launched the Third Blackfyre Rebellion. Of the deeds done then, both good and ill—of the leadership of Maekar, the actions of Aerion Brightflame, the courage of Maekar's youngest son, and the second duel between Bloodraven and Bittersteel—we know well. The pretender Haegon I Blackfyre died in the aftermath of battle, slain treacherously after he had given up his sword, but Ser Aegor Rivers, Bittersteel, was taken alive and returned to the Red Keep in chains.

If interested: Aerion Brightflame's ill act was killing Haegon Blackfyre

So many things could have happened (Bittersteel still has it/stole it on the way to the wall, Aerion took it, Haegon died with it, etc.) But I like the idea of Bloodraven/Bittersteel dueling again with Blackfyre/Dark Sister.

The Future

• GRRM has so many ways he could go with these swords. We know early drafts of Tyrion's early chapters in ADWD mentioned the word "sword" as a possible gift from Illyrio to Young Griff before it was later removed from the chapter (along with a ton of other Blackfyre stuff according to Elio Garcia)
• So they really could wind up anywhere. Dark Sister (if its in the cave) could go to Meera or Jon (if he gives Jorah Longclaw). Also GRRM has confirmed that Longclaw is NOT Dark Sister or Blackfyre.
• With Euron as a potential former student of Bloodraven (its possible Bloodraven gave Euron the dragon egg he uses to pay the FM). Since we know Euron has a valyrian steel suit of armor, does he have a sword to go with it?

Something else potentially relevant for the future:

Did Tohbo Mott ever teach Gendry the secrets of reworking Valyrian steel?

GRRM: Interesting question. -SSM: Valyria-related Subjects: 1 Jan 2002

It also should be noted that with Bloodraven still alive, and now the Golden Company (Bittersteel) how that plays out.

TLDR: I would love to see Blackfyre and Dark Sister clash for a potential third time, and there are so many possibilities.