Missing info from: Seasons 3 and 4
Individual episodes of Person of Interest generally feature a standalone storyline following the episode's person of interest. In addition, a series of character and story arcs that span multiple episodes are woven through the season's episodes.
Active Story Arcs
Harold Finch, Nathan Ingram and the Development of the Machine
Subplot: Nathan Ingram's Death
Nathan Ingram was a close friend of Finch and a co-creator of The Machine. He is revealed to die under mysterious circumstances in 2010 and it is soon implied his death is connected with The Machine. The truth is finally revealed at the end of Season 2: He was murdered because he planned to go public with The Machine, and his death was camouflaged as a terrorist attack.
Main article: Nathan Ingram
Subplot: The Location of the Machine and The Program
A major subplot involves the search for the Machine itself. Many people were looking for The Machine, including Nathan Ingram and Root. At one point, Finch claimed he doesn't know its location, but at the end of the second season, he does. (It is unclear if he lied on the first occasion or he determined in later.) Nathan Ingram got very close - he narrowed the search to 3 possible locations. The location was finally found at the end of the second season, but The Machine has moved itself away shortly before.
In the episode “Relevance”, it is revealed that someone is killing off all the people who know the location of The Machine (although they may not be aware of its true nature).
This story arc overlaps with Root story arc, but other people search for it, too. Before the Season 2 finale, the Machine moved itself from its original location, leaving Decima, the Program personnel, Root and other groups attempting to determine its location.
Main article: The Machine
Carl Elias and the New Mafia
From the beginning of the series (“Mission Creep”), Reese and Finch regularly encounter a mysterious criminal boss, known only as Elias. He is one of the most powerful criminals in NYC, and is trying to take control of all organized crime in the city.
This story arc was apparently resolved in the episode “Flesh and Blood”, when Elias was arrested and incarcerated at Rikers Island, but he remains powerful even in jail; his influence continues to be felt, sometimes even helping Finch and Reese.
He is later found to be living in a basement along with his lieutenant Scarface. He is assisting Carter by giving her the location of Maksim, a pawn shop owner who a person of interest was to meet with. He offers to take down the Russian Mob and HR, but Carter walks away. However, Elias secretly orders Scarface to take the diamonds and money from Maksim. He also later helps Carter by explaining one of HR's money laundering systems.
Eventually most of Elias' men are killed and he is forced into hiding by Samaritan.
Main article: Carl Elias
Root, whose real name is Samantha Groves, is a violent hacker with an interest in 'freeing' The Machine. She first appeared as a hacker for hire in “Root Cause”.After Finch and Reese foiled her, she became obsessed with how they did it. She eventually deduces the existence of The Machine, and tries to free it multiple times. This has included kidnapping Finch, and later, enlisting his assistance. Though by season's end she is foiled again, she receives a call by The Machine in the cliffhanger.
Main article: Root
At the end of “Trojan Horse”, Finch informs Reese that a Chinese company, Decima Technologies, was responsible for the virus that would ultimately cause The Machine to fail. Later explained by Root in “Zero Day”, Decima hired assassins were staking out every pay phone in Manhattan, with the intent of gaining 24 hour access to the Machine. They fail, but it is left unclear if they will try again. After learning the existence of a similar system - Samaritan, a project like the Machine made by Arthur Claypool - Decima successfully stole some drives, restoring its data and bring it online in a 24 hours test period. Using Samaritan, despite failing to eliminate Finch's associates (John Reese and Sameen Shaw) due to Root's interference , Decima did manage to track down and kidnap Finch's former fiancee Grace Hendricks in exchange for him. Finch agreed to their offer and was captured. However, Reese, Shaw and Root got a hold of the servers used by Decima to power Samaritan. Main article: Kara Stanton Main article: Greer Main article: Decima Technologies Main article: Samaritan
Subplot: The Ordos Laptop
In “Dead Reckoning”, it is revealed that the laptop was given to the Chinese by Harold Finch and it contained parts of The Machine's code, it was used to create a virus to infect The Machine.
Finch was aware that sooner or later, someone would try to attack The Machine. So he created a "virus within a virus" that would allow the perpetrator to attack it, but at the same time, modified it so it could defend itself.
Main article: The Ordos Laptop
Subplot: Samaritan's quest to eliminate its rivals
Main article: Martine Rousseau
Subplot: Samaritan employs technical genius from all over the world
Main article: Claire Mahoney
Subplot: Samaritan gains control of the government
Subplot: Team Machine's struggle to hide from Samaritan and adjust to their new life while still protecting Persons of Interest
Completed Story Arcs
Search For the Man in the Suit
Nicholas Donnelly was an FBI agent who was one of the lead investigators into the "man in the suit", a man who authorities believed was terrorizing the city but was actually just John Reese. The FBI was unaware of his identity.
Mark Snow was a CIA agent with similar intentions, wanting to track down John Reese and who knew that he was in fact the man in the suit. John worked with Snow in the CIA previously, and after attempting to kill John during one of his missions in Ordos, wanted to find John (seemingly to finish the job).
Eventually, Donnelly managed to track John down at a bank in New York City. When the FBI visited the sight where he was believed to be, they saw four men, all dressed in suits and were unable to determine who was the man they were looking for. After some work from Carter and Finch, John's identity seemed to check out and he was released from Rikers Penitentiary, where he was being held.
After being released from prison, however, John met with Carter and Donnelly discovered that John was in fact the man in the suit and that Carter worked with him as well. Donnelly arrested both of them on the spot and brought them into his car, planning to bring them to a safehouse and deal with the authorities later. During the ride, however, Kara Stanton, another one of Reese's ex-partners, crashed a truck into the vehicle and killed Donnelly, sedating John and taking him captive.
After a long cat-and-mouse game between Kara, John, and Snow ensued, the FBI agent replacing Donnelly believed that Snow was in fact the man in the suit and after Snow and Kara's death, closed the FBI's investigation.
A group of corrupt cops connected with high politicians, called "HR" (Human Resources) was a regular opponent of Reese and Finch . After achieving many little victories, Reese and Finch, with help from a vengeance driven Carter, took down HR and it's head, Alonzo Quinn. However, this victory didn't come free, as it resulted in Carter's death.
Also, Fusco, a former member of HR, helped to take down Carter's killer: Patrick Simmons.
Main article: HR