Jack Harkness's vortex manipulator

Jack Harkness possessed a vortex manipulator.

He used it during his time as a Time Agent, and beyond.

History
To escape from the Lect, Jack used his manipulator to travel from Fluren's World into the Doctor's TARDIS, bypassing its security systems. (COMIC: Weapons of Past Destruction)

After Rose Tyler as the Bad Wolf resurrected Jack following his extermination by the Daleks, (TV: The Parting of the Ways) Jack used his manipulator to leave 200,100. He attempted to reach 21st century Earth, but missed and ended up in 1869. The manipulator burnt out after this trip. However, Jack was still able to use at least some of the other functions besides time travelling and teleporting. (TV: Utopia)

In 1927, Jack used the manipulator to forge a visa for Angelo Colasanto. (TV: Immortal Sins)

By 2007, Jack was able to control the Torchwood Hub’s invisible lift with the manipulator. (TV: Everything Changes)

When Jack, the Tenth Doctor and Martha Jones were being attacked by Futurekind on Malcassairo in the year 100,000,000,000,000, the Doctor repaired the vortex manipulator with his sonic screwdriver, transporting them to Earth in 2008. The group used it to teleport to the Valiant, from which Martha subsequently escaped, also by use of the manipulator. (TV: The Sound of Drums) Martha considered the vortex manipulator in her Top Five Worst Ways To Travel because it made her sinuses and organs ache and caused her throat to become bloody. (PROSE: The Story of Martha)  took it from her when she was captured and later also used it to teleport with the Doctor from the Valiant to Earth, where he threatened to destroy the planet. The Doctor was able to use the manipulator to teleport them both back to the Valiant. Once the Master was defeated and Jack made his way back to the Torchwood Hub, the Doctor deactivated the manipulator, saying that he couldn't leave Jack with a time-travelling teleport. "You could go anywhere in time, twice; second time to apologise", according to the Doctor. (TV: Last of the Time Lords)

When he was in Cardiff, Captain John Hart sent messages to Jack’s manipulator. (TV: Kiss Kiss, Bang Bang, Fragments)

When the Daleks relocated Earth to the Medusa Cascade and invaded it, Jack used information from the UNIT Project Indigo (based on Sontaran teleport pod technology) to reactivate the teleport function on his vortex manipulator (he still couldn't time travel). He then teleported to find the Doctor, arriving just after the Doctor was shot by a Dalek, which Jack then killed. (TV: The Stolen Earth) After the Daleks were defeated, the Doctor deactivated the manipulator again, sternly saying "I told you - no teleport". (TV: Journey's End)

Using the Rift Manipulator, Jack was briefly able to recharge his vortex manipulator, enough to shunt Andy Davidson to 1919 and back. (PROSE: They Keep Killing Andy)

Jack was wearing the manipulator when a bomb placed in his stomach was detonated, destroying the Torchwood Hub. (TV: Children of Earth: Day One) The manipulator survived, though it’s strap was lost, and it was retrieved from the ruins by Gwen Cooper. Her husband Rhys procured a new strap for it and she gave it back to Jack, who used it to signal a cold fusion freighter on the edge of the Sol system to transmat him aboard. (TV: Children of Earth: Day Five)

Jack was wearing the manipulator when he returned to Earth in 2011 during Miracle Day. It was briefly confiscated by Rex Matheson during Jack and Gwen’s rendition to the United States. (TV: Rendition)

Eventually, Jack bequeathed his vortex manipulator to UNIT after one of his deaths. It was held in their Black Archive under the tightest secrecy, with UNIT keeping it hidden its allies (particularly the Americans). The Doctor was aware that UNIT had the device but refused to give them the activation code. When the Eleventh Doctor found himself locked in the Tower of London in 1562, he inscribed the vortex manipulator's activation code on a stone pillar so UNIT could find it in 2013. Upon receiving the code, Clara Oswald activated the vortex manipulator and used it to travel to 1562 so she could rescue the Doctor. (TV: The Day of the Doctor)

By 2021, Jack had reacquired his vortex manipulator somehow and repaired at the very least the teleport function. While breaking the Thirteenth Doctor out of prison, he somehow managed to smuggle it in with him and stashed it in a hidden compartment in the floor of one of the cells of the prison. After rescuing the Doctor, Jack led her to the vortex manipulator and used it to teleport himself and the Doctor back to her TARDIS. When the Doctor asked how Jack had managed to smuggle the vortex manipulator in, he asked "you really want me to answer that?" Jack later used the vortex manipulator to transport himself, Ryan Sinclair and Graham O'Brien to a Dalek saucer in order to destroy it. On Yasmin Khan's signal, Jack teleported the group and Jack Robertson out seconds before the Daleks could exterminate them while Ryan detonated the bombs and destroyed the Dalek ship. Unlike her predecessor, the Thirteenth Doctor allowed Jack to retain full use of his vortex manipulator when he departed. (TV: Revolution of the Daleks)

Behind the scenes
Jack's vortex manipulator was used much in the same way as the Doctor's sonic screwdriver in the context of Torchwood, being capable of doing just about whatever the writers wanted it to. Its only known limitation, within the context of Torchwood episodes, was being unable to time travel or teleport, both functions having been deactivated by the Tenth Doctor.

Although a vortex manipulator first appeared in the TV story The Empty Child, "vortex manipulator" was not identified as the name of the time agents' time travel device until the TV story Human Nature (where it was described, but not shown), and Jack's portable wrist strap was not directly described as one until the TV story Utopia.