The Franky Family is a group of ship dismantlers and bounty hunters, brought together under the leadership of Cutty Flam (better known as Franky). They are the outcasts of Water 7, those whose dream was to work for Iceburg in the Galley-La Corporation, but failed the entry exam to become shipwrights. As a result, they had nowhere else to turn, other than to Franky (who also considers himself something of a failure). Though they do cause some trouble to the citizens of Water 7, they are good at heart. They protected Water 7 from other looters. The entire Franky Family are totally inept at saving money for a long period of time and on more than one occasion wasted millions of hard-earned (sometimes stolen) beli for the sake of spending (though their leader is not exactly a market planner himself).
After feeling guilt for the incrimination of his master Tom, Cutty Flam wandered around the alleys and backstreets of Water 7, surviving off of what he could find and beating up other low-lifes. He adopted the name Franky after attacking his first victim.
Water 7 Arc and Enies Lobby Arc
During the starting chapters of the Water 7 Arc, they first encounter the Straw Hat Pirates, and beat up Usopp in order to steal the money he was carrying for Nami and Luffy. Usopp pursues the Franky Family and confronts them for the money, only to be severely beaten once more. When Luffy finds out it was the Franky Family who committed the act, he, Zoro, Sanji and Chopper singlehandedly defeat the entire group (except for Franky and the Square Sisters, who were off the island), and destroy the Franky House, their hideout. Clamoring for revenge, Franky would attempt to get revenge on Luffy, only for the three-way battle between him, Luffy, and the Galley-La Company foremen to end in a draw.
Later on in the story, the group encounters the CP9 agency and are forced to reveal their second hideout, where Franky is hidden. When Franky is kidnapped, the Franky Family forms an alliance with the Straw Hat Pirates (who are chasing after Nico Robin), in order to rescue him. This turns out to be a wise move as once on Enies Lobby, the Judicial Island, the Frankies help fight off the countless naval marines and government agents that come after them. Their King Bulls, Sodom and Gomorrah, prove instrumental in helping them make their way across the island to the entrance to the main island. They then rush to escape Enies Lobby before they are annihilated by the Buster Call, while Franky himself is helping Nico Robin to secure an escape ship for the Straw Hats.
The Galley-La foremen, Franky Family, Oimo, and Kashi reached the Main Gate only to be attacked by three Buster Call ships there. The Marines announced that they are all dead, but they actually survived, with Oimo and Kashi blocking the brunt of the assault, while Paulie prevented them from falling into the bottom of the waterfall with a web formed by his unbreakable rope. They are last seen leaving Enies Lobby by the Sea Train with the Galley-La shipwrights.
Post-Enies Lobby Arc
Two days later after the Enies Lobby incident, the Franky Family are seen rebuilding the Franky House (noting a poor rebuilding job). Sodom and Gomorah are seen recovering from their injuries. Later they discover that their actions at Enies Lobby have been overlooked by the World Government (most likely due to Aokiji pulling some strings). However, Franky, they discover, was not so lucky, and has been given a bounty of 44,000,000. They immediately rush to the Straw Hat Pirates to show them their new bounties as well as Franky's and ask them to take him with them as he is a wanted man and it is too dangerous for him to stay at Water 7.
They also inform them that Franky is the son of a pirate. To their great relief they find that Luffy already was planning to ask him to join the crew. As the Straw Hats made their way to Scrap Island, the Franky Family then started to head to the Franky House (having asked Mozu and Kiwi not to come because of what they planned to do) to find Franky there and not at Scrap Island like they thought. Tamagon then stole his speedo in order to make him chase them back to Scrap Island. Along the way they kept passing his speedo around to each other while they passed through the streets of Water 7, all the while, Franky was attempting to get it back. They almost failed if not for the untimely arrival of Monkey D. Luffy to the fray which eventually led to Franky to be shot out of a cannon onto Scrap Island.
The Franky Family then arrived and gave him his bag to in preparation for his journey. This enraged Franky because they were telling him what to do. They responded by saying that they cared about his happiness too which moved him to the point of tears (though he hid it) and he finally agreed to join. They then found that Usopp was not on board. After Usopp rejoins and the Straw Hats escape, the Franky Familly try to decide what to do next but cannot think of anything. Iceburg then shows up and offers some of them jobs to which they start calling him "new big bro", much to his chagrin.
In the anime, they were shown reading about Luffy's return to Marineford and were glad that he was alright.
Two Years Later
After the timeskip, Mozu and Kiwi became bartenders in Blueno's old bar. Zambai and some other members have joined Galley-La and formed the Zambai's Company Union.
The Franky House stands at the northeast coast of Water 7 and is the place where the Franky Family lives and functions as their base. Like most of Water 7, it is very colorful. The house has big gold letters reading "FRANKY HOUSE" in front and has large robotic-esque "arms" hanging off the sides.
The house is ultimately destroyed by the Straw Hats after the Franky Family stole most of the Straw Hat's Beli and Usopp, blaming himself, tried, and failed to get the money back by fighting them alone. It is eventually rebuilt, albeit poorly, sometime after the Enies Lobby arc by the rest of the Franky Family while Franky works on the Thousand Sunny.