Warning, Spoilers!
(possible book and up until episode 5)

Who is Azor Ahai?

Azor Ahai, or the Prince who was promised, is the knight chosen by the Lord of Light to defeat the White Walkers and bring back the dawn. He carries a flaming sword named Lightbringer. The sword is forged after two failures (where the sword breaks) by plunging the sword into the heart of his wife Nissa Nissa. Her pain ignites the sword to flames and creates Lightbringer.

Azor Ahai is who Melisandre is obsessed with finding throughout the show, she first decides Stannis is Azor Ahai, now she think's it's Jon Snow. Azor Ahai is basically like a “reincarnation” of the knight who sent the White Walkers packing last time they tried to conquer Westeros. He wields a fiery sword and “The Darkness Flees” before him. Also, according to Missandei, the incorrect translation of the prophecy means Azor Ahai could also be a princess, which includes our silver haired dragon rider in the list of potential candidates. Here's the candidates and some of my thoughts:

Jon has had the most traditional heroes arc throughout Game of Thrones, making him the hottest contender for Azor Ahai. Since we have found out he is the son of Lyanna Stark and Rhaegar Targaryen, he contains both the blood of ice (Stark) and fire (Targaryen). Many think this explains the name of the novels, and it explains his affinity with both the Night's King, and Drogon the dragon, as shown in the latest episode of Game of Thrones. He was also successfully resurrected by Melisandre, suggesting that he is important to the plans of the Lord of Light. He has successfully killed a White Walker with his Valyrian steel sword, which could be forged into Lightbringer.

So far so good. The only downside to this theory is that George RR Martin tends to be critical of the heroes journey of the perfect, flawless saviour, and most of the characters of Game of Thrones are morally ambiguous. Daenerys wants to free slaves and support the common people but also quite happily burns people alive and hangs them alive from crosses. Jaime Lannister shoved a kid out of a window and has sex with his sister but saved Brienne twice, nearly sacrificed himself trying to kill Daenerys and has shown all the signs of a morally complex character. Theon has turned from a child killing traitor to a pitiable character. Ramsey Bolton...oh wait. Anyway, Jon is such an obvious choice for Azor Ahai that it feels a little out of place in GRRM's way of storytelling. Jon's popularity makes him the most obvious red herring.

Also, Melisandre has a history of being wrong about who Azor Ahai actually is.

Daenerys literally rides her “flaming sword” into battle, a giant black dragon named Drogon. Who better to conquer the king of Ice and the dead but a fiery living queen on the back of a giant fire breathing lizard? Red Priestesses across the Narrow Sea have already suggested that Dany is Azor Ahai, the Princess who was Promised. She has an incredible story. First she is sold by her brother to a nomadic warrior; Khal Drogo, with whom she quickly becomes an equal after a short period of being repeatedly raped by him. She is then is reborn in the fire after Drogo's death with three dragons and then conquers many cities, freeing slaves and creating a massive army of dedicated, loyal followers who will happily impale themselves on swords for her. Daenerys was also born on Dragonstone, fulfilling a part of the prophecy saying the Prince would be born amid salt (on a small island in the sea), and smoke (she is metaphorically reborn in the flames).

Daenerys is also an obvious choice. Again, perhaps too obvious. The “Daenerys is Azor Ahai” theory is nearly as predictable as the Jon Snow theory, we were only thrown off the scent for a while with the use of male pronouns to refer to Ahor Azai. She is the other possible “red herring” to fool viewers/readers into believing she is Azor Ahai. She is painted as somewhat of a saviour in the show and it just seems all too neat.

There's also the possible Targaryen madness. Although to me, Cersei is the most obvious representation of a Mad Queen, Daenerys has frequent temper tantrums in which she ignores her advisors. Her face when she was riding Drogon and burning men alive last week looked disturbingly gleeful. There is definitely a hint of the Targaryen madness about her, even if she turns out to be an overall force for good by the end. Her potential madness has not gone unnoticed by Tyrion and Varys too.

I absolutely love this theory, although I don't think it's true. Someone pointed out on Reddit that mistranslations of Azor Ahai point to Jaime. Azor Ahai, according to this theory doesn't mean “Lord+Light” but “Gold+Hand”. Well that's specific, isn't it? Perhaps our golden handed antihero is not such a bad choice after all. Other signs that point to Jaime are the idea that his sacrifice of his hand to prevent Brienne being raped was the start of his redemption story, and the “Valonqar” prophecy. Remember the flashback where a young Cersei went to see a witch in the woods, who told her she would have three children who would all die, and she would then be killed by a younger brother? She always thought it was going to be Tyrion and so hated and mistrusted him. But Jaime is the younger twin and seems far more likely. With Cersei going all Mad Queen, Jaime might just have to end her. This would fulfill the “Nissa Nissa” part of the Azor Ahai prophecy.

The biggest con of this theory to me is that if Jaime is Azor Ahai, then he has a long way to go before he takes up his flaming sword. He is fighting on the opposite side from Jon Snow who is trying to fight the White Walkers, appears still to be very much in love with Cersei and is now the father of her unborn baby. While I do think Jaime will kill Cersei, I'm not sure I buy him as the Lord of Light, however I do think it would be a much more interesting route for the show to take than the obvious choice of Dany or Jon.

This one centres on the theory that the sword LightBringer is a metaphor not an actual sword. What if the translations are taken all too literally and the bringer of the light is not a sword but a person? What if Jon Snow is the one who brings back the light, perhaps by riding on the back of a dragon? So Jon is himself the sword. Remember Nissa Nissa? In this interpretation of events, Catelyn Stark; Bran's mother, is Nissa Nissa. A lot of people think Bran messing around in the stream of time led him to manipulate events to shape the ultimate battle into what it's actually going to be. This is a pretty complex one involving Bran actually being Bran the Builder who built the wall, and also being the voices whispering to Aerys Targaryen, telling him to “Burn them all” meaning the White Walkers. Before he gains full control of his powers, Bran could make a number of mistakes, or perhaps the Mad King's actions are necessary to bring forth Daenerys as the dragon Queen and Jon Snow as Lightbringer, to beat back the advancing dark.

The biggest issue I have with this one is that it's complicated. Such a theory might do well in the books where GRRM has hundreds of pages to go on at length and describe Bran's actions, but I really doubt it's going to translate into the 8 or 9 more episodes that are left in season 7 and 8.

I also think that time travel is a trope used all to often for convenience in too many stories. I feel it would a bit of a cliche to use Bran's time travelling abilities in this manner. I find this one the least likely choice but still a really interesting theory nonetheless.

So many Lords of Light!