Superphone

"Superphone" was the nickname Rose Tyler gave to her mobile phone after it was altered by the Ninth Doctor, giving it the ability to communicate through time and space. (TV: World War Three) Similar "superphones" were created by the Doctor for many of their companions after Rose. The Doctor sometimes had one of their own.

Abilities
A superphone could be used to make telephone calls through time and space. It could also work in places where there was no phone service.

The Ninth Doctor altered Rose's mobile phone into a superphone by adding a chip into it, enabling Rose to call her mother Jackie while on her adventures with the Doctor. Rose had experienced distress at the thought of going to a time period where her mother would quite likely be long dead by then, which prompted the Doctor to modify her mobile to contact Jackie in real time elapsed from the point she entered the TARDIS. (TV: The End of the World)

According to the Ninth Doctor, the TARDIS itself also boosted mobile signals significantly. (PROSE: Rose)

The Tenth Doctor altered Martha Jones' phone into a superphone, this time using only his sonic screwdriver rather than a technological addition to the phone, imparting it with "Universal Roaming" service. (TV: 42)

Donna Noble's phone was enhanced by the Doctor as well. (TV: The Doctor's Daughter)

The Doctor altered Barclay's phone to call UNIT. (TV: Planet of the Dead)

The Doctor also added Universal Roaming to Gabby Gonzalez's phone using the sonic screwdriver. He told her messages could be sent to the screwdriver or the psychic paper. In this particular upgrade, Gabby received a "Doctor" app on her phone through which the messages could be sent. (COMIC: The Fountains of Forever)

After having her brand new smart phone wrecked by the Wire, the Tenth Doctor made it up to Alice Wu by upgrading her old mobile phone giving it a direct line to the TARDIS. (PROSE: Loose Wire)

There were at least two superphones available for use in the Eleventh Doctor's TARDIS: Amy Pond's phone and a flip phone occasionally used by the Doctor. (TV: Day of the Moon, The Doctor's Wife) Rory Williams also possessed a superphone. (TV: Dinosaurs on a Spaceship)

Examples of use
During the Slitheen crisis, Jackie Tyler called her daughter. Rose's superphone was the only phone working in the sealed Cabinet Room. It provided an outside line for the Ninth Doctor to communicate with Mickey Smith and Jackie and coordinate efforts against the Slitheen. Using the connection, the two groups figured out the Slitheen's weakness. The Doctor talked Mickey through shooting a Harpoon missile at 10 Downing Street, destroying the Slitheen. Later, Rose received a call on it from the Doctor in the TARDIS. (TV: World War Three)

Abuse of the superphone
Adam Mitchell used Rose's superphone to phone his mother in his own time to send knowledge of the future back to his time for his own gain. Knowing the potential damage he could do to the natural course of history, coupled with an experience where Adam's underhanded behaviour during a journey to Satellite Five nearly got everyone killed, the Doctor expelled him from the TARDIS. Furthermore, he used the sonic screwdriver to overload and obliterate the phone in his mother's house to ensure the messages he left on its answering machine would be irrecoverable. (TV: The Long Game)

Other uses
While stuck on Pete's World, Rose's superphone still worked. She used it to find out about that world. During the invasion of the Battersea Power Station, the Tenth Doctor had Mickey find a much needed code. He transmitted it to Rose's phone via text. The Doctor plugged the phone into a nearby console and the code shut down the Cybermen's inhibitors, defeating them. The Doctor had Mickey keep Rose's superphone, as it had the code he needed programmed into it. (TV: The Age of Steel)

Rose received another superphone by the time she was stranded on Krop Tor. This phone had no signal there, after the TARDIS was briefly lost, but was intercepted by the Beast, who sent a message saying, "He is awake." (TV: The Impossible Planet)

While stuck on a ship falling into a sun, Martha's phone was modified by the Tenth Doctor to call her mother. During her travels with the Doctor, she did this a lot, unaware her calls were being monitored. When she needed to answer a trivia question, Martha called her mother to look it up on the Internet. (TV: 42)

Following her ordeal during "the Year That Never Was", Martha Jones gave her phone to the Doctor to contact him should he ever be needed. (TV: Last of the Time Lords) Martha called him, bringing him back to Earth to assist with UNIT's investigation of the ATMOS facility. (TV: The Sontaran Stratagem)

The Doctor later modified Donna Noble's phone to call Martha Jones. (TV: The Doctor's Daughter)

Martha's phone was also used by Torchwood, Mr Smith and Harriet Jones, all using the Cardiff rift to contact the Doctor when Earth was teleported to the Medusa Cascade by Davros. (TV: The Stolen Earth)

When he was stuck on San Helios, the Doctor modified Barclay's phone into a superphone to communicate with UNIT on Earth. The Doctor used it to communicate with Doctor Malcolm Taylor and Captain Erisa Magambo to find a way back and deal with the threat of the stingrays. Even though his phone was unmodified, Nathan called the Doctor on the superphone, showing that any phone could call a superphone, no matter where it was. The Doctor brought the people on the 200 home. He used the phone to tell Malcom to close the wormhole despite being hung up on twice. What happened to it afterwards is unknown. (TV: Planet of the Dead)

Apparently Lady Christina de Souza either learned of the Doctor's superphone number or obtained Barclay's phone, as at one point the Doctor answered his superphone in the TARDIS thinking it was Christina calling him. It was actually Martha. (COMIC: Tesseract)

When the Doctor briefly suffered a retro-regeneration, his psychic paper sent a telepathic feedback to Gabby's superphone, allowing her to access the messages on it and locate the Doctor. (COMIC: The Fountains of Forever)

Canton Everett Delaware III used Amy Pond's mobile phone to receive a call from the Eleventh Doctor's flip phone in 1969, before cell networks existed, indicating that both phones must have been upgraded by the Doctor at some point. (TV: Day of the Moon)

The Doctor used Amy's phone to keep in touch with her when she and Rory were trapped in the TARDIS. (TV: The Doctor's Wife)

Clara Oswald's phone was capable of video calling to and from the TARDIS. (TV: Before the Flood)

Strengths and weaknesses
A superphone had the ability to call the person the caller was looking for no matter where in time or space they were, referred to as "free roaming". It locked onto the user's relative time. It worked in areas with no service, as proven when the Ninth Doctor, Rose, and Harriet Jones used Rose's superphone to communicate with Jackie Tyler and Mickey Smith while trapped in the cabinet room, which should have been impossible. (TV: World War Three)

Rose could not get any reception on the planet Krop Tor, but was startled when she received a call foreshadowing the Beast's escape. (TV: The Impossible Planet) Martha also had trouble contacting the Doctor when the Earth was forced out of sync with the rest of the universe. She found her phone couldn't contact the superphone. As Martha herself said, this was probably because the Daleks were blocking all interstellar transmissions. Wilf and Sylvia could not contact Donna's superphone with their normal phone and Rose Tyler's superphone was also non-functional. (TV: The Stolen Earth)

Rory Williams once found a superphone on the TARDIS and used it to access the Internet. Because the TARDIS had no firewalls, Rory's dabbling resulted in the TARDIS being flooded by holographic spam. (COMIC: Spam Filtered)

Despite being able to communicate across time, Clara's phone was still unable to get a signal when taken inside the Faraday cage at the mining facility called The Drum. (TV: Before the Flood)

Precursors
The Second Doctor modified Brigadier Lethbridge-Stewart's mobile radio, boosting it through the TARDIS communication systems to function through the TARDIS' forcefield, though it lacked the range to work after the TARDIS and the surrounding laboratory had been taken to the anti-matter universe. (TV: The Three Doctors)

The Fifth Doctor and Nyssa created an experimental device called a temporal interocitor to communicate across spacetime. This device was implied to be a prototype for the superphone. (AUDIO: Renaissance of the Daleks)

Behind the scenes
The Doctor has to date been shown three times on screen turning their companion's mobile into a "superphone": Rose Tyler's in TV: The End of the World, Martha Jones' quite late in her first season, in TV: 42, and Donna Noble in TV: The Doctor's Daughter. With Amy Pond and Clara Oswald, phone upgrades were simply taken as a given, with Clara seen using two distinctly different models of phone during her time.

In series 11, Yasmin Khan explicitly regains mobile reception only upon returning to 21st century Earth in TV: Arachnids in the UK.