Police

Police forces were government organisations charged with the responsibility of maintaining law and order.

The Fourth Doctor compared the Krynoid pods to police as both travelled in pairs. (DW: The Seeds of Doom)

The Judoon and Atraxi were police forces working throughout Mutter's Spiral, though Earth's planetary development level in the 21st century placed it out of their jurisdiction. (DW: Smith and Jones, The Eleventh Hour, SJA: Prisoner of the Judoon) In the 26th century the term "police" was considered old-fashioned (NSA: Prisoner of the Daleks)

Instances of Police appearance

19th century

A number of police and detectives were involved in the cases surrounding the suicide of Gordon Seavers and the murders at Banquo Manor. (EDA: The Banquo Legacy)

20th century

In 1927, New York police shot Jack Harkness and arrested Angelo Colasanto. (TW: Immortal Sins)
To make sure that Elizabeth II's coronation in 1953 went smoothly, the police hid away people whose faces were consumed by the Wire. (DW: The Idiot's Lantern)
A lone policeman patrolled the Coal Hill area of London in 1963. When passing a junkyard owned by I.M. Foreman, he did not notice the presence of a strange, humming police box. (DW: An Unearthly Child)
Thanks to the intercession of the temporarily miniaturised First Doctor and his companions, police (including PC Bert Rowse) were able to arrest corrupt industrialist Forester for the murder of Arnold Farrow. (DW: Planet of Giants)
Pursued by the Daleks through time and space, the Doctor's TARDIS materialised in the yard of a police station in 1965 Liverpool, its scanner malfunctioning. After some trouble with the local police, the First Doctor managed to repair the malfunctioning scanner and depart. (DW: The Daleks' Master Plan)
When a policeman approached the TARDIS, the First Doctor hung a sign reading "out of order" on the front of his ship. The policeman saw the sign and moved on. (DW: The War Machines)
A policeman spotted the Second Doctor and his companions on a runway at Gatwick Airport and chased after them. When they scattered, the policeman went after Ben. (DW: The Faceless Ones)
Two Autons impersonated police in a failed attempt to kill the Third Doctor and Jo Grant. (DW: Terror of the Autons)
A policeman witnessed Brigadier Lethbridge-Stewart and the Third Doctor's high-speed chase of Lupton. (DW: Planet of the Spiders)
Gustave Lytton had two robot henchmen which resembled policemen. (DW: Resurrection of the Daleks/Attack of the Cybermen)
In 1999, the Eighth Doctor stole a motorbike from a policeman in order to reach the beryllium clock in San Francisco. (DW: Doctor Who)

21st century

In 2006, the police were interviewing Rose Tyler and the Ninth Doctor when Rose had disappeared for a year (but only a few days from her perspective). In the interim, Mickey had been questioned multiple times for murder. Sip Fel-Fotch Passameer-Day Slitheen had disguised himself as police, ready to kill anyone associated with the Doctor. (DW: Aliens of London/World War Three)
On Christmas 2006, a policeman was reporting on the people with A+ blood ready to jump from roofs of London thanks to the Sycorax's blood control. (DW: The Christmas Invasion)
In 2007, Sally Sparrow approached DI Billy Shipton, who had been investigating disappearances at Wester Drumlins. (DW: Blink)
Gwen Cooper, before she worked for Torchwood Three, had been a police officer, along with most of her friends. Shortly before Gwen joined Torchwood, she and Andy Davidson looked into John Tucker's murder and were involved with stopping a fight at a pub. (TW: Everything Changes)
Police were called to investigate Matt Stevens' unusual death. (TW: Day One)
Mark Goodson had himself arrested by police for protection from the fairies. (TW: Small Worlds)
After Torchwood Three's encounter with the Brynblaidd villagers, the villagers were arrested by police. (TW: Countrycide)
Police cordoned off a 200-year-old corpse of a soldier near an Arcateenian transporter. (TW: Greeks Bearing Gifts)
DI Henderson looked into the prevented murder of Carol and Danny. (TW: Greeks Bearing Gifts)
Kathy Swanson's team investigated Max Tresilian's murder of Mark and Sarah Briscoe and Alex Arwyn. Her team later helped Torchwood find the right code to break the Torchwood Hub's lockdown. (TW: They Keep Killing Suzie)
Police had a violent encounter with English Civil War soldiers when they came through the Cardiff Space-Time Rift. (TW: End of Days)
In 2008, Sarah Jane Smith was arrested for the kidnapping of "Ashley Stafford" (who Mr Smith had faked evidence to make him appear as the son of Jay and Heidi Stafford). Sarah Jane's connections with UNIT had her quickly released. (SJA: The Lost Boy)
The Torchwood Three team pretended to be police while questioning Rhys Williams. (TW: Meat)
Following a fight, Jack Harkness and the newly-undead Owen Harper were arrested. (TW: Dead Man Walking)
Rift activity caused several senior police officers at Andy Davidson's police station to be mauled by Weevils. (TW: Exit Wounds)
On Easter 2009, DI Macmillan was on the case for arresting Lady Christina de Souza for robbery. The Tenth Doctor pretended to arrest himself inside a police box for allowing her escape. (DW: Planet of the Dead)
Following Miracle Day in 2011, the Welsh police helped Rex Matheson to extradite Jack Harkness and Gwen Cooper to the United States. (TW: The New World)
A team of armed police, led by Andy Davidson, rescued Gwen Cooper's family from being held hostage. (TW: Immortal Sins)
Ralph Finch discovered the category 1 Geraint Cooper hiding in Swansea, and subsequently took him away to Cowbridge Overflow Camp. (TW: The Gathering)
Rory Williams was mistaken for a plain-clothes policeman by Ambrose Northover in 2020 Cwmtaff. (DW: The Hungry Earth)
Harry Pike, father of Darius Pike, was one of the last policemen to serve in England, before The Department replaced them with CCPCs. (K9TV: The Last Precinct)

5,000,000,000 A.D.

In 5,000,000,023, The New New York Police Department arrested the Sisters of Plenitude after the NNYPD were alerted of experimenting on "the flesh". (DW: New Earth) In 5,000,000,029, the department, and all police authority on New Earth were consumed by the Bliss virus. (DW: Gridlock)

Individual police officers

Humans

Metropolitan Police Service

Behind the scenes

The policeman seen at the beginning of An Unearthly Child has the distinction of being the very first character to appear on Doctor Who.

Apocrypha

Tom Campbell was a Police Constable from London who witnessed a robbery and burst into Dr. Who's TARDIS. (Film: Daleks' Invasion Earth 2150 A.D.)