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Earlier in the 1960s
Hamato Yoshi is a member of the foot clan. He becomes entangled in a love triangle with Oroku Nagi over Tang Shen.
1969
Hamato Yoshi moved from Japan to New York in order to escape the bitter love triangle between himself and Oroku Nagi. Yoshi became homeless in order to avoid detection. While living in the sewers with the rats as his friend, Yoshi one day found four turtles, recently bought from a pet store by an unnamed boy who accidentally dropped them in the sewer. The four Turtles were covered with a strange glowing ooze . The substance caused the turtles - most recently exposed to Yoshi - to become humanoid, while Yoshi - most recently exposed to sewer rats - became a humanoid rat, and started going by the pseudonym "Splinter".
Splinter choose their names from a book about Renaissance artists: Donatello, Leonardo, Michelangelo, and Raphael.
1969-1984
Splinter raises them from infancy as his sons and trains them in the art of ninjutsu. Meanwhile, Oroku Saki has left Japan and tracked Yoshi to New York City, where he intends to destroy him once and for all. He has become associated with Krang, a disembodied alien brain who has been banished from his home, Dimension X, where he was a great warlord. Saki has taken on a new persona, donning a suit covered with razor spikes, complimented by a long cape, and a metal mask over his mouth. He has also taken on the pseudonym "The Shredder".
It becomes clear that the mutagen that transformed the Turtles and Splinter into their new forms was dumped into the sewer by Shredder in an effort to destroy Yoshi. Shredder thought it was a deadly poison. The Turtles vow to take revenge on the Shredder for dishonoring their master, as well as turning him into a rat. The Turtles want to force him to turn Splinter back into a human again. This quickly evolves into stopping Shredder's ongoing criminal career with the aid of Channel 6 reporter April O'Neil. The Turtles quickly take on the role of vigilante crime-fighters operating outside of the jurisdiction of law enforcement against any criminals. The Turtles are constantly preoccupied with hiding their existence. Burne Thompson, April's employer, and Vernon Fenwick, a Channel 6 cameraman frequently blame them for the trouble that the Shredder and Krang cause.
1987-1993
Casey Jones takes up vigilantism. Shredder, Krang, Bebop and Rocksteady (two street thugs morphed into animal form by exposure to the Shredder's mutagen), and a small army of robotic Foot Soldiers try to destroy the Turtles and take over the world. Much of their quest for world domination hinges on bringing the Technodrome (Krang's mobile fortress, and his and The Shredder's base of operation) to the surface as it was stuck in the earth's core.
1993-1995
The Turtles challenge Shredder and the Foot Clan on a roof top and defeat them. Shredder falls off the roof, holding a bomb that blows up.
Shredder, returned from the dead, seeks his revenge on the Turtles. In an ambush on Christmas Eve, the Turtles, Splinter, April O'Neil, and Casey Jones are forced to retreat to Casey's grandmother's farmhouse in Northhampton, Massachusetts. Leonardo is badly hurt in the battle, losing a hand, and replacing it with one of his Katana. Raphael also loses an eye.
A year later, the Turtles return to defeat the Shredder. In a dramatic conflict, Shredder reveals to Leonardo how he was brought back to life. After the first conflict, the Foot recovered his remains. Using a combination of mysticism and science, his body was revived as a colony of worms. Leonardo decapitates the Shredder.
New York is the site of the battle between the factions of the leaderless Foot Clan battle each other for dominance. In response to the chaos, the Japanese Foot Clan sends Karai to be the new leader. Karai enlists the help of the Turtles to restore order. In exchange for the help, the Foot Clan must never bother the Turtles again.
1996-1999
The Turtles decide to go their separate ways. Meanwhile, Baxter Stockman constructs a robotic body while imprisoned at DARPA. He places his brain in it and starts his mission to get revenge on April. Raphael encounters Stockman and they battle. Raphael is thrown from the roof and captured by DARPA.
The three other Turtles reunite and defeat Stockman. Having defeated him, they turn their attention to their lost brother. With the help of Casey, they infiltrate DARPA and find several animal hybrid species, including Triceraton, being experimented on. The Turtles battle DARPA. A freed Triceraton threatens a planetary invasion from his colony. The Turtles escape with the help of another test subject.
1999-2001
The Turtles' birthday party is interrupted by cyborg assassins, Pimiko, and her 'ninja babies'. Donatello and Raphael are both badly injured during the fight. Both Donatello and Splinter are kidnapped. The remaining Turtles relocate to a graveyard while Donatello and Splinter are taken to upstate New York by helicopter. Donatello and a cyborg fall from the helicopter, still fighting. Donatello kills the assassin with its own gun, but breaks his shell and is paralyzed. Donatello fuses with the cyborg. Being cyborg, Donatello gains many new abilities, but loses his shell.
Raphael and Michelangelo are ambushed by Pimiko and her ninja babies at the graveyard while Leonardo is in a trance. Raphael, Leonardo, and Michelangelo find the spot where Donatello fell. Seeing only his shell, they believe him dead. They set off on a rescue/revenge mission. The three Turtles fight Mako and find that Donatello is alive, though he is a cyborg. Reunited, they break into Lord Komodo's lair. Lord Komodo turns himself into a Komodo Dragon.
2002-2013
Turtles, are living together in their sewer lair beneath New York City. April O'Neil and Casey Jones have been married for some time and remain in contact with the Turtles from their nearby apartment. Splinter lives at the Northampton farmhouse, where he has become a "grandfather" of sorts to Casey's teenage daughter, Shadow. Unbeknownst to them, Karai had actually been sent to look into Shredder's death. She was able to recover his remains and turned to Eric Sacks at TCRI to see what he could do. He regenerated Shredder in a vat. and developed a cybernetic suit to maintain his vitals.
Shredder knowing first hand the value of the mutagen seeks to bring about a new world and hatches a plot to release the mutagen in gas form across New York City. Michaelangelo is visiting Northampton recovering from an injury when Raphael, Leonardo, and Donatello discover this plot almost by accident as they are fighting an increase in street crimes. Donatello uses his cyber-abilities to hack the security systems while Leonardo and Raphael go to Sacks Tower in an attempt to foil it. Leois thrown from the roof and Raph goes to save him, but shredder lassos Raph's neck. Leo died on the streets and Raph died hanged. The foot discovers Donatello's location and he barely escapes.
When he gets back to safety, Donatello informs Michaelangelo of the events and they vow to seek vengeance for their brothers. Up until that point, only the Turtles had been known to the people, and even then, most were skeptical. The release of the 'strange green cloud' caused a massive upsurge in hybridization and animal hybrids became publically known.
2015
Donny and Mike had moved to an abandoned pizzaria to train when they were sought out by other hybrids, at which point they secretly opened up the area as a dojo for hybrids to fight Shredder and the Foot. Those who knew about the dojo only found out by word of mouth, and very few pure humans were permitted into it (Major exceptions being the Jones family obviously)