Superphone

The superphone was the nickname Rose Tyler gave to her mobile phone after it was altered by the the Ninth Doctor, giving it the ability to communicate through time and space. Similar "superphones" were created by the Doctor for many of his companions after Rose. The Doctor had one of his own.

Abilities
The 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 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. (DW: The End of the World) Adam Mitchell used it to phone his mother in his own time to send knowledge of the future back to his time for his own gain. The Doctor expelled him from the TARDIS for this. (DW: The Long Game)

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. (DW: 42)

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

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

There were at least two superphones available for use in the Eleventh Doctor's TARDIS: Amy's phone and a flip phone occasionally used by the Doctor.

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 missile at 10 Downing Street, destroying the Slitheen. Later Rose received a call on it from the Doctor in the TARDIS. (DW: World War Three)

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. (DW: The Age of Steel)

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. (DW: 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. Martha called him, bringing him back to Earth to assist with UNIT's investigation of the ATMOS facility. (DW: Last of the Time Lords, The Sontaran Stratagem)

The Doctor later modified Donna Noble's phone to call Martha Jones. (DW: 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. (DW: 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. (DW: 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. (IDW: Tesseract)

Canton Delaware used Amy Pond's cell 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.

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

Strengths and weaknesses
The 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 the user's relative time. The only exception to this last was the Doctor himself. 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. (DW: 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. (DW: 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. (DW: The Stolen Earth)

When the Doctor modified Donna's phone, he called Martha on it despite her being on a different planet and the phone she used being an ordinary cell phone. When he used Barclay's modified phone, Nathan called him despite being on the same planet as he was. This indicates that any phone could call the superphone or could be called by it, no matter if it had service or not.

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. (IDW: Spam Filtered)

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. (DW: The Three Doctors)

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