|Although this article is based on canonical information, the actual name of this subject is pure conjecture.
- "What wife would do the things I've done for you? What wife would trust you the way I've trusted you? When you gave me those drops and told me to pour them into Jon's wine. My husband's wine. And you told me to write a letter to Cat telling her it was the Lannisters..."
- ―Lysa Arryn explains her reasons for killing her husband.
Following Robert's Rebellion, Jon Arryn was appointed as Hand of the King to King Robert. While Robert devoted his energies to whoring, drinking and hunting, Arryn was effectively left to govern the realm and apparently did so wisely. After seventeen years as Hand, Arryn died suddenly of a fever.
At Winterfell, Eddard Stark learns of Jon Arryn's death and that King Robert and the royal party are bound for Winterfell. Eddard realizes that Robert intends to name him as his new Hand. He considers refusing but Lady Catelyn receives a letter from her sister Lysa, Jon's widow, claiming that her husband was poisoned by the Lannisters. With this news, Eddard is convinced to accept Robert's offer so he can investigate Jon's death.
Once in King's Landing, Eddard starts his investigation. He discovers that just prior to his death, Arryn acquired a book from Grand Maester Pycelle: The Lineages and Histories of the Great Houses of the Seven Kingdoms and visited several of King Robert's bastard offspring. He deduces that Arryn was investigating the parentage of Robert and his wife Cersei's three children. Varys, the Master of Whisperers, warns Eddard that he believes Jon was actually poisoned using tears of Lys, a poison that is rare, costly and difficult to detect. When Eddard inquires as to motive for the murder, Varys simply replies, "He started asking questions".
Tyrion Lannister is apprehended by Lady Catelyn, who accuses him of trying to kill her son Bran. She takes him to the Eyrie, Lysa's home. Once there, Lysa accuses him of orchestrating Jon Arryn's death too. After a trial by combat, Tyrion is acquitted.
Under interrogation from Tyrion Lannister, Grand Maester Pycelle admits that Arryn had discovered that truth about Cersei's children; that they were not fathered by Robert, but by her brother Jaime, and he intended to inform the King. Pycelle also admits that Arryn was poisoned but strongly denies doing it himself. Tyrion nevertheless says Pycelle was complicit and let Arryn die because of the threat he posed to the Crown.
The full truth of Arryn's death is finally revealed by Lysa Arryn in a conversation with her new husband Petyr Baelish. To prove her love for him, she reminds Baelish that, on his instructions, she laced Jon's wine with Tears of Lys and then sent a letter to her sister falsely accusing the Lannisters for this murder.
Eventually, Baelish is forced to answer for his multiple crimes, including Jon's murder, by Sansa Stark, who proceeds to have him executed by her sister Arya Stark. Sansa discovered this fact from her omniscient brother Bran Stark. 
In the books
In the A Song of Ice and Fire novels, the assassination of Jon Arryn is carried out in the same manner: Tears of Lys on his wine. However, Lysa Arryn is not only motivated by her obsession with Littlefinger but also her refusal to have her son Robin sent off to Dragonstone to be fostered by Stannis Baratheon, as Jon Arryn intended. Although this plan was totally in keeping with expectations for noble boys in Westeros, Lysa's mental instability and overprotectiveness of her son meant she would have none of it. Lord Arryn was being treated by his own Maester Colemon, who was going to give him purges to get the poison out, but because Pycelle knew that Arryn knew the secret of Cersei's incest, he sent the man away and just gave Lord Arryn painkillers to ease his passing. It must be noted that Pycelle never knew who actually poisoned Arryn, and his later explanation of his actions to Tyrion was truthful.
Jon's death prompts Stannis, who had initiated the investigation about Robert's bastards and enlisted Arryn to help him (while in the TV show it was Arryn alone who conducted the investigation), to flee to Dragonstone, believing he may be the next target - never realizing that Jon's murder wasn't originally about Cersei's children.
At one point in the first novel, Ned examines the Valyrian steel dagger, trying to figure why anyone would want Bran dead; he has a strong gut feeling that the dagger and Bran's fall are somehow linked to Arryn's murder. Ned's assumption is wrong: while there is a very vague connection between Bran's fall and Arryn's death (the link is Jaime and Cersei's incestuous relationship), the failed attempt on Bran's life has absolutely nothing to do with Arryn's death.
In "A Storm of Swords", Lysa blurts out the whole truth about Arryn's death when she drags Sansa to the Moon Door. Littlefinger tries in vain to hush Lysa, feeling uncomfortable that the secret is revealed in the presence of Sansa and Marillion. The series presented this as separate events, with Lysa blurting out the truth within earshot of a Septon, later incoherently raging to Sansa who took a while to piece it all together, and finally implying the truth in the Moon Door scene.
It is not revealed in the novels exactly why Littlefinger wanted Jon dead, but there are several logical possibilities:
- To make Lysa free for marriage and to gain control over the Vale by marrying her
- To prevent Jon from telling Robert Baratheon the truth about Cersei Lannister's children
- To turn the Starks and Lannisters against each other via the letter that Lysa sent to Catelyn, in which she falsely put the blame on the Lannisters
So far in the books, Jon Arryn's true killer is not yet common knowledge, known only to four people, and only two of them are known to still be alive - Littlefinger and Sansa.