The R+L=J theory claims Jon Snow most probably is the son of crown prince Rhaegar Targaryen and Ned's sister Lyanna Stark.
The Tower of the Hand has an excellent analysis of this theory:
Jon Snow's Parents
And Westeros' Citadel also provides a summary:
Jon Snow's Parents
A Wiki of Ice and Fire:
Jon Snow Theories
Radio Westeros podcast:
A Dragon, a Wolf and a Rose
Frequently Asked Questions:
How can Jon be a Targaryen if ordinary fire burned his hand?
Targaryens are not immune to fire. It's a myth that has been refuted by a list of Targaryens being burned. Danaerys 'the unburnt' was indeed unscathed when she hatched the dragon eggs, but that has not stopped her being burned on other occasions. See this thread on Targaryen fire immunity.
Don't all Targaryens have hallmark Valryian silver-golden hair and purple eyes?
Not all of them: Valarr and Queen Alysanne had blue eyes. Bittersteel, who like Jon was half first men blood, had brown hair. Baelor Breakspear and his son(s) and Jon's own half-sister Rhaenys had the Dornish look (dark hair, black eyes, olive skin). Rhaenyra Targaryen's three sons all had brown hair and brown eyes even though both their parents had light silver-gold hair.
Had Jon Valyrian features, it would give his parentage away: "He had the Stark face if not the name: long, solemn, guarded, a face that gave nothing away. Whoever his mother had been, she had left little of herself in her son." Tyrion got the bit about the mother wrong, though: his mother was the Stark.
If Jon isn't Ned's son, then why does he look so much like him?
Jon looks very like Arya, and Arya looks very like Lyanna. Jon is Ned's nephew, and Lyanna and Ned looked similar.
Ned is too honourable to lie. If he says Jon is his son, doesn't that mean he must be?
Ned tells Arya that sometimes lies can be honourable. His final words, a confession of his guilt, are a lie to protect Sansa. While a lie can be honourable, cheating on his wife isn't, so Ned's famed honour points to Jon not being his son.
How can Jon be half-Targaryen and have a direwolf?
He's also half Stark, through Lyanna. Ned's trueborn children are half Tully and that doesn't stop them having direwolves.
Why doesn't Ned ever think about Lyanna being Jon's mother?
Ned doesn't think about anyone being Jon's mother. If he did, there would be no mystery. He names 'Wylla' to Robert, but we do not see him thinking of Wylla being Jon's mother.
There's a hidden hint at who Jon's mother might be: In chapter 4, Eddard's internal monologue goes "Lyanna ... Ned had loved her with all his heart." and in chapter 6, Catelyn thinks "Whoever Jon's mother had been, Ned must have loved her fiercely".
Why would Ned not at least tell Catelyn?
We don't have a list of what Ned promised to Lyanna, but know he takes his promises seriously. Maybe he promised not to tell anyone. In Chapter 45, Ned is uncertain what Cat would do if it came to Jon's life over that of her own children. If Catelyn knew that Jon was Rhaegar's son, she might feel that keeping him at Winterfell presented a serious risk to her own children. Ultimately, Catelyn did not need to know, so maybe Ned simply chose to be on the safe side.
Doesn't Ned refer to Robb and Jon as "my sons in the very first chapter?
In speech, not in thought. Ned is keeping Jon's parentage secret. He never thinks of Jon as his son: In Chapter 45, Ned thinks of his children "Robb and Sansa and Arya and Bran and Rickon and explicitly excludes Jon from the list. ADwD Chapter 34 has Bran's vision of younger Ned in the Winterfell godswood: "...let them grow up close as brothers, with only love between them," he prayed, "and let my lady wife find it in her heart to forgive..." which not make sense if they are brothers.
Since Rhaegar was already married, wouldn't Jon still be a bastard?
He might, or might not. There was a tradition of polygamy among Targaryens in the past, so the possibility that Rhaegar and Lyanna married is not easily ruled out. A pro-legitimacy argument is this: The presence of the three kingsguards at the Tower of Joy is best explained if they were defending the heir to the throne, which Jon would only be if he was legitimate.
Can we be certain polygamy is not illegal?
Aegon I and Maegor I practised polygamy. In Westeros, unlike a constitutional monarchy, royals are not subject to the law. So if there ever was a law against it, it did not apply to the Targaryens: In Chapter 33 it says "like their dragons the Targaryens answered to neither gods nor men". Examples demonstrate that it was considered an option for Targaryens: Aegon IV and Daemon Blackfyre may have considered it for Daemon, Jorah Mormont suggested it to Daenerys as a viable option, and she said the same about Quentyn Martell.
George R.R. Martin says in this SSM: "If you have a dragon, you can have as many wives as you want". There is alsothis SSM predating the worldbook.
On Polygamy essay by Ygrain with additions by Rhaenys_Targaryen
Weren't the Kingsguard at Tower of Joy on the basis of an order from Aerys, to guard Lyanna as a hostage?
If so, why would they have apparently made no effort to use this leverage against Robert and Ned? Some argue their Kingsguard vows would have taken precedence and still have required them to leave the Tower to protect Viserys when he became heir -- unless there was another that took precedence [Jon]. Others think they were guarding Lyanna as a hostage at the Tower of Joy. Some say that makes little sense: She would better be kept hostage at King's Landing, and wouldn't require kingsguards to guard her. The mere presence of three kingsguards implies something more important: guarding members of the royal family or maybe the heir.
Frequently suggested readings: At the tower of joy by MtnLion and support of the toj analysis by Ygrain
Isn't there an SSM that says the 3 Kingsguard were following Rhaegar's orders though?
The SSM you may be thinking of is probably this: The King's Guards don't get to make up their own orders. They serve the king, they protect the king and the royal family, but they're also bound to obey their orders, and if Prince Rhaegar gave them a certain order, they would do that. They can't say, "No we don't like that order, we'll do something else."
We know from Barristan, protecting the king is the first and most important of all kingsguard duties. Jamie suggests some other KG to stay with the king when he wants to leave for the Trident and we also learn of a ritual that is performed when all KG meet and the king is guarded by someone who is not from the order.
"Protect vs Obey" is an ongoing subject of debate that is unlikely to be settled until we know more. Either viewpoint is compatible with R+L=J.
Wouldn't Viserys take precedence anyway? Rhaegar died without becoming king, and doesn't the world book call Viserys, not Aegon, Aerys' new heir?
No, in the case of an eldest son dying before the king dies, a grandson comes before a younger son. Even in the case the grandson is yet unborn at the time of death, he would succeed (heir apparent vs. heir presumptive). The world book is written with a Lannister bias (it may be propaganda to undermine Dornish support for the Targaryens) and in hindsightby maesters who have never learned all of what we know from Ned's dreams and memories. If it still turns out to be true... see the next answer.
Are matters of succession just as clear as presented here?
Succession quarrels are a part of medieval power play and even a very clear inheritance could well be contested. So maybe in King's Landing things did happen as the world book says. Rhaegar and Aerys may have been at odds over the succession. Rhaegar told Jaime before leaving for the Trident that he intended to call a council, and The Great Councils of the past have dealt with matters of succession. Who would have accepted such a change is a question worth asking.
Ned is dead. Who's going to tell anyone about it?
Bloodraven and Bran may have learned of it through the weirwood network. Benjen might know. Checkov's CrannogmanHowland Reed is the sole survivor of the encounter at the Tower of Joy, and George R.R. Martin has stated he has not yet appeared because he knows too much about the central mystery of the book. "They had found him [Ned] still holding her [Lyanna's] body" tells that there also was someone else besides Howland to find Ned.
Why is this important? What impact can it have on the story?
The careful way the mystery of Jon's parentage was created is reason to believe it's important. What impact it will have on the rest of the series is still unknown.
This theory is too obvious and too many people believe it to be fact. How can it be true?
It is not so obvious to the majority of readers. Some will get it on their first read, but most will not. Readers who go to online fan forums, such as this, still represent a very small minority of the readership. Also, A Game of Thrones has been out since 1996. That's more than 18 years of readers being able to piece together this mystery. Crowd-sourced internet-based mystery solving like this inevitably make solved mysteries seem more obvious in hindsight.
George R.R. Martin is a "breaker of tropes, there can be no hidden prince, it's simply too cliché.
In order to break a trope it needs to be installed in the first place. It is yet unknown what will happen to Jon in the future. Being the son of Lyanna and Rhaegar does not imply the fairy-tale style happy ending associated with the hidden prince trope.
Is there a list of all R+L=J clues that have been found?
There is a list of R+L=J hints, clues and foreshadowing compiled by sj4iy. (the link is currently inactive)
Since this theory has been refined so well, will Martin change the outcome of the story to surprise his fans?
He has stated that he won't change the outcome of the story just because some people have put together all the clues and solved the puzzle.
A thread for discussing strengths and weaknesses of the theory that Jon Snow's parents are Rhaegar and Lyanna.
Please click on the spoiler below to reveal links to all previous editions of this thread