The Public Surveillance Unit (or PSU) is the branch of the Justice Department tasked with monitoring the civilian population via a wide network of cameras. They were notorious at their height for being able to find out anything about a given target
While it's not known how long PSU has existed for, the ambitious (some would say power obsessed) Judge Jura Edgar was a member in the 2090s and used her position to help several Judges get away with murdering suspects. 
From 2100 to 2121, Edgar ran PSU and honed it into one of the most powerful agencies in the city - as well as a way to have power for herself.
After the Statue of Judgement was rebuilt in 2117, the PSU headquarters were moved into it. By 2120, PSU had uncovered minor criminality and embarrassing dirt on almost the entire civilian Council and Mayor of Mega-City One for purposes of control.  When Dredd undermined this operation, Edgar used the Unit's resources and connections to enact revenge by finding dirt on him and Galen DeMarco.
PSU was a key objective of Nero Narcos in the Second Robot War and he swiftly took control of the Statue. Many Judges questioned how Narcos had pulled off his attack without PSU noticing. Following Edgar's subsequent reassignment as warder of a Cursed Earth Labour Camp, leadership of the Unit was handed over to Judge Rog Niles, formally of the SJS.
In 2134 the Statue was destroyed by Sov agents as part of the Day of Chaos plot. A destructive riot and looting spree immediately broke out as the public were aware that, without the PSU, the Judges would struggle to arrest anyone. Without the Unit, the Judges were unable to stop the Day of Chaos - Dredd would note that Justice Department had become too complacent on its surveillance might.
Afterwards, Hershey planned to fold Public Surveillance Unit, the Wally Squad, and the Special Judicial Squad into the new Undercover Operations Division. (The merger doesn't appear to have happened.) A later plan had surveillance equipment placed in rebuilt homes in order to better spy on people, an operation that had be shut down after being exposed.
When the Statue of Judgement was rebuilt once more in 3138, PSU moved back in.
- While the PSU wasn't formally created until "The Cal Files", a prototype version called the "Control Centre" appears in prog 6 by Malcolm Shaw and Mike McMahon.