Mobile phone

Mobile phones were personal communicators. (AUDIO: Absent Friends) Americans sometimes called them cell phones, or just cells. (TV: Arachnids in the UK)

Earth culture
On the planet Earth, they became popular in the early 21st century. (TV: Rose, World War Three, Smith and Jones) They were crucial elements of teenagers' social lives. After realising that Martin Trueman had stolen his mobile and was now among the stars, Clyde Langer exclaimed, "I'm nobody. I'm no one! It's a social disaster." (TV: Secrets of the Stars)

Interference
When Jack Harkness rescued Rose Tyler in his ship's tractor beam, he told her that her mobile phone interfered with his instrument. Rose responded that "no one ever believes that." (TV: The Empty Child) Toby Silverman and Celeste Rivers also took the mobiles of Sarah Jane Smith, Rani Chandra and Clyde Langer, also claiming that they interfered with the equipment. (TV: The Eternity Trap) Ryan Sinclair displayed similar horror when all his online accounts were deleted when the Thirteenth Doctor reformatted his phone to track the signal linking the DNA bombs to the Gathering coil. (TV: The Woman Who Fell to Earth)

The Doctor and mobile phones
The Ninth Doctor had his own cellphone on one occasion. (TV: Boom Town) The Tenth Doctor later kept another phone given to him by Martha Jones. (TV: Last of the Time Lords) The Twelfth Doctor, during his travels with Clara Oswald, began carrying a cellphone. (TV: The Zygon Invasion, Extremis)

The Doctor also lent Elvis a mobile phone at some point before their thirteenth incarnation. By 1 December 1955, Elvis let Frank Sinatra borrow his mobile, against the Doctor's wishes. (TV: Rosa)

Companions and mobile phones
During her travels with the Doctor in her youth, Sarah Jane Smith was exposed to mobile phones but he refused to give her one, despite her wishes, claiming it was anachronistic. During her later years, she sardonically remarked that, despite such a claim, he had proceeded to gift her a robot dog from the year 5000. (COMIC: Vortex Butterflies)

In October 1987, Serena Paget was surprised that Melanie Bush had a mobile phone as they were not common at that time. (AUDIO: We Are The Daleks)

Anji Kapoor carried her mobile with her when she began travelling with the Eighth Doctor, and was even able to use it when she travelled into her future. (PROSE: Escape Velocity, Trading Futures)

The Eighth Doctor set up mobile phone-like devices for companions Anji and Fitz Kreiner when they spent a selective amount of time on certain planets. (PROSE: The Book of the Still)

Rose Tyler attempted to use her mobile phone from 2005 when she was in 1987, bewildering Pete Tyler. (TV: Father's Day) When Jack Harkness first met, and rescued, Rose, he asked her to turn off her phone as it was interfering with his tractor beam's ability to safely compute a descent pattern. (TV: The Empty Child)

Sarah Jane's teenage friends used mobile phones to keep in contact as well. This resulted in their phones being destroyed by their enemies on many occasions, with Sarah Jane's adoptive son, Luke Smith, having gone through seven phones alone in two years. (TV: Invasion of the Bane, Warriors of Kudlak, Whatever Happened to Sarah Jane?, The Last Sontaran, The Day of the Clown, The Mark of the Berserker, The Mad Woman in the Attic, The Curse of Clyde Langer, The Man Who Never Was)

Superphones
Rose Tyler, Martha Jones and Donna Noble had their mobiles adjusted by the Doctor to "universal roaming"; (TV: 42) Rose by the Ninth Doctor, (TV: The End of the World) and Martha (TV: 42) and Donna (TV: The Doctor's Daughter) by the Tenth Doctor. These were called superphones. (TV: World War Three) They could call nearly anywhere in time and space (TV: The End of the World, 42) and work in areas with no service, such as the Cabinet Room. (TV: World War Three) However, Rose was unable to phone her mother Jackie Tyler from Krop Tor, which the Doctor described as being five-hundred years from Earth. (TV: The Impossible Planet) The Tenth Doctor also modified Barclay's phone in this manner so he could communicate with UNIT. (TV: Planet of the Dead)

Martha's superphone was left with the Tenth Doctor. (TV: Last of the Time Lords) later being integrated into the TARDIS following its repairs. (TV: The Eleventh Hour)

Later incarnations of the Doctor also similarly modified the phones of others who travelled with them:

Clara Oswald used her mobile on the Orient Express to call Danny Pink on 21st century Earth. (TV: Mummy on the Orient Express) She subsequently used it to contact the Twelfth Doctor - both audio and video calling - on numerous occasions. (TV: Dark Water, Before the Flood, et al.)

Courtney Woods was able to use her mobile phone to post to Tumblr in her own era while aboard the Twelfth Doctor's TARDIS in the year 2049. (TV: Kill the Moon)

The "superphone" modification was a refinement of the temporal interocitor, a communication device invented by the Fifth Doctor and Nyssa. (AUDIO: Renaissance of the Daleks)

Later alternatives
By 2050, mobile phones appeared to have been replaced by the vid-com. (TV: Liberation, The Cambridge Spy, Angel of the North)

Behind the scenes
Once treated in the series as a major event akin to being entrusted with a TARDIS key (Rose Tyler, Martha Jones), the upgrading of a companion's phone to superphone status is now often taken as a given in BBC Wales Doctor Who; the evident upgrading of Clara's phone, for example, has never been depicted on screen.