The identity of Azor Ahai is a heavily disputed one. Listed here are some of the most popular ones.

Azor Ahai is a Single Character

Stannis Baratheon

Stannis Baratheon was reborn amongst salt (the sea) and smoke (fires) on Dragonstone. He also wields a sword forged in the fires, that he calls Lightbringer. He claims descent from Rhaella Targaryen and is currently aware of the threat of the Others.

Maester Aemon Targaryen doesn't believe Stannis' sword to be Lightbringer as it does not exude any heat.

This theory is held by many, including Melisandre and Ser Godry Farring.

Jon Snow

When Melisandre looks into her fire to see visions of Azor Ahai she sees nothing but snow. She thinks it may be ash, but is aware that some connection to Jon may be present. Jon himself has a dream of him fighting the Others with a red sword in his hands. Jon's sacrifice of Ygritte was similar to the sacrifice Azor Ahai made of Nissa Nissa in the legend.

Jon Snow may have the blood of the dragon, as a popular fan theory is that he is the son of Rhaegar Targaryen and Lyanna Stark. His wounds smoke when his brothers betray him, and Bowen Marsh cries. the body of Ser Patrek of King's Mountain (whose sigil is a star) lies bloody above him.

Daenerys Targaryen

The original prophecy made about Azor Ahai's rebirth was made using the Valyrian term for dragon, which is gender neutral. As such, the reborn Azor Ahai could be either male or female. Daenerys was first born on Dragonstone getting the salt from there, and was then reborn in the funeral pyre of Khal Drogo, hence the smoke, and there her three dragons were born of petrified eggs, hence dragons born from stone.

Supporters of Daenerys being Azor Ahai include Maester Aemon Targaryen and High Priest Benerro

Victarion Greyjoy

Whilst travelling to Daenerys Targaryen Victarion has his sword wound healed by the red priest Moqorro, who claims that he sees a long future for Victarion in service to R'hllor. After healing the wound the arm smokes and they are at sea. Victarion is also known to have murdered his wife, something the original Azor Ahai had to do.