The Cobblepot crime family was a powerful criminal organization in Gotham City founded by Oswald Chesterfield Cobblepot, A.K.A. "The Penguin." It was founded in the wake of the Falcone-Maroni war and quickly emerged as the dominant power in Gotham's underworld, seizing control of both remnants of Maroni and Falcone's organisations.
The family suffered two major but temporary downfalls. The first came after Oswald was hunted down by the GCPD for the murder of Theo Galavan, going into hiding after his release from Arkham, and once again after the combined efforts of Edward Nygma and Barbara Kean. Penguin retook control of the criminal underworld once again after Barbara's disappearance and his defeat of Nygma, opening the Iceberg Lounge in place of the The Sirens and issuing licenses for crime in Gotham. With Oswald incarcerated for the murder of Martin, the crime family is currently inactive.
After the retirement of Don Carmine Falcone, and the deaths of Sal Maroni and Fish Mooney, Oswald Cobblepot stepped forward to fill the power vacuum created by the gang-lords' absence. He quickly took control of Gotham's criminal underworld and began running the city alongside former associates of both Falcone and Maroni, dubbing himself the "king of Gotham."
Penguin along with Tommy Bones (who formerly worked for Sal Maroni), Butch Gilzean (who formerly worked for Fish Mooney), and Victor Zsasz (who formerly worked for Carmine Falcone) collected debts owed to retired Falcone, as, having taken over his organization, Penguin declared that the debts were now owed to him. Selina Kyle was in his presence where Cobblepot stated to Jim Gordon that he keeps her around like a a pet cat.
Cobblepot later called a meeting with subordinates, as the recent exploits of Jerome Valeska and The Maniax were causing their business interests to suffer. As such, Oswald ordered that all violent acts from then on would require his permission, to "restore faith" in their power. Oswald and his group were later dragged into Theo Galavan's scheme when he revealed that he had Gertrud Kapelput in his clutches.
The Penguin committed the murders of the mayoral candidates that opposed Theo's and had Butch hire the arsonists The Pike Brothers former Fish Mooney loyalists to carry out an arson based robbery on a Wayne Enterprises building thus discovering the knife which would play a role in the temporary fall of his empire.
After Tabitha Galavan "reprogrammed" Butch Gilzean and stabbed Gertrud in front of Oswald, this leads to Oswald retaliating by orchestrating an attack on Theo's post-election party where he used look-a-likes of him to fool the police.
Oswald later rallied his gang and Selina Kyle when they help Jim Gordon, Harvey Bullock, and Alfred Pennyworth when it came to rescuing Bruce Wayne from the Order of St. Dumas after he was captured by Theo Galavan. The group found themselves in a gunfight which resulted in the members of the Order ending up dead.
Oswald and his gang went into hiding and Oswald planned for them to use a disused meat packing plant as their hideout. However they were attacked by Victor Fries, who began to freeze his underlings and chase him down. Oswald managed to escape with the help of Selina Kyle and a surviving underling, who drove Oswald away to safety. But when Oswald went to Gordon and Bullock for help, he was taken outside of Gotham and told to leave town for good as Captain Nathaniel Barnes was after him. Despite this warning, Oswald returned to Gotham.
Oswald's empire fell in his absence and he was a wanted man for the murder of Theo. After spending about a month living on the streets of Gotham, he was eventually arrested and brought to the precinct by Barnes and later sent to Arkham Asylum after pleading insanity. Butch succeeded Oswald as the new crime lord of the Gotham underworld. 
Three months later Oswald formed an alliance to kill a resurrected Theo Galavan. Afterwards Oswald appeared to be back in control and Butch took orders for him, such as attacking a prison bus seemingly containing Hugo Strange - a chief psychiatrist from Arkham Asylum who had tortured Oswald during his time in there.
Six months after the escape of the Indian Hill prisoners, Oswald appears to have taken control of the Gotham Underworld with Butch as his willing right hand man. Offering Barbara to run protection for her Sirens Nightclub as it is built on his land to which she refuses his offer preferring to run things herself along Tabitha.
After being elected as Mayor of Gotham Penguin effectively rules Gotham as kingpin of it's criminal underworld and as the legally elected Mayor of the City. Whilst continuing the rule Gotham, jealousy pushed him to arrange the death of Edward Nygma's girlfriend. Eventually Nygma discovered the truth and, along with Barbara Kean, orchestrated the fall of Penguin's criminal empire as well as his reputation as Mayor. After Nygma and his co-conspirators had finally destroyed everything Oswald had built, he shot his former friend in the gut and dumped him in river.
Oswald survived the attempted murder and was nursed back to health by Ivy Pepper. She subsequently helped him to reorganize his syndicate and take control of Gotham once more. With the aid of former Indian Hill inmates Victor Fries and Bridgit Pike, he began planning his re-ascension as well as his revenge on Edward Nygma.
After another failed attempt on Penguin's life, Nygma brought him to the same harbor where he had first attempted to kill him, only for Oswald to reveal that his former friend had fallen into a trap. Penguin delivered the final blow of the conflict by having Fries freeze the Riddler alive. He kept the resulting ice block as a reminder to never again let love weaken him.
Returning to his mansion with Ivy, Cobblepot announced the idea for a new club which he dubbed "The Iceberg Lounge" noting the fact that is corresponds with his nickname "Penguin" but also because of it's main attraction being the cryogenically frozen Edward Nygma being on display.
In the three months following this, Penguin gained influence in Gotham City and has unionized crime, in which criminals have to come to him for a license before they commit a crime so he can get a cut of the take. Penguin helps to cut down the crime rates, and manages to get control of the Gotham City Police Department as his policies have prevented attacks on the department and the rate of cop murders has gone down.
- Oswald Cobblepot (boss; incarcerated)
- Butch Gilzean (underboss; defected)
- Gabe (bodyguard, defected; prior to his death)
- Armand (bodyguard, defected; prior to his death)
- Edward Nygma (chief of staff; defected)
- Stretch (bodyguard; fate unknown)
- Ivy Pepper (enforcer and co-leader; defected)
- The Sirens
- Victor Zsasz (enforcer; defected)
- Mr. Penn (bookkeeper; fate unknown)
- Victor Fries (enforcer)
- Bridgit Pike (enforcer; fate unknown)
- The Dentist (torture expert; defected)