In Doctor Who, the Doctor more often than not travels with a companion in the TARDIS. He seems to enjoy having someone to bounce ideas off of and to help him out of the inevitable sticky situations they end up in. There have been many famous and beloved companions during the history of the show, from Ace to Rose to the Ponds, but there has only been one Time Lady.
Romana joined the Fourth Doctor on his adventures and definitely gave him a run for his money, keeping him on his toes and challenging him on more than one level. Not only was she a great companion on the show, but there's never been another like her.
Romanadvoratrelundar, or Romana for short, made her debut appearance in Doctor Who in the 1978 episode The Ribos Operation. Played by Mary Tamm, Romana was a Time Lady from Gallifrey who initially joined The Doctor at the request of the White Guardian. In season 16 of the show, the White Guardian and the Black Guardian featured prominently, with the White Guardian the personification of order and the Black Guardian his opposite representing chaos.
The main storyline of season 16 concerned assembling the Key to Time and, saw Romana join the Doctor on his quest despite being less than impressed with him. Romana began her tenure on the show as a stuck-up and distant character, believing herself intellectually better than the Doctor — a far cry from humans who are often awestruck by the time-traveling alien. Romana was a highly clever individual, stubborn and strong-willed. However, as they traveled together, she did begin to appreciate the Doctor's sense of adventure and warmed up to him somewhat.
As the course of the season progressed, actress Mary Tamm didn't appreciate the way in which Romana's character devolved from a capable and intelligent woman to a damsel-in-distress style character. This ultimately led to her decision not to return for further episodes past season 16. However, this wasn't the end for Romana, enter Lalla Ward and Romana II.
When the show returned for season 17, Mary Tamm did not reprise her role, but Romana was back. At the start of "Destiny of the Daleks," Romana is seen in the TARDIS essentially trying on new bodies to see which she likes best. In a twist on the regeneration that audiences were used to, which are usually serious moments, Romana was able to change to an appearance that she found suitable. That appearance happened to be the form of Princess Astra, a character that Lalla Ward (Romana's new actress) had played at the tail end of season 16. Romana has thus also been one of very few companions to be played by multiple actors, and the only one shown to regenerate.
After her regeneration, this second incarnation of Romana became known as Romana II. As she continued her adventures with the Fourth Doctor, Romana II noticeably mimicked his style and also acquired her own Sonic Screwdriver, a design that the Doctor coveted and tried to swap for his own. After her regeneration, Romana and the Doctor also had a much closer and friendlier relationship, although this could have been informed by Lalla Ward and Tom Baker's real-life romance that began behind the scenes.
As they traveled together, it became clear that this Romana was a more effervescent and outgoing version of the original Romana, although still retaining some of her initial superior attitude. During their travels, Romana II dons a school girl's outfit (apparently Ward's idea) and frequently engages in sarcastic and snarky banter with The Doctor, each trying to one-up the other. She's hardly the only companion to keep the Doctor on his toes, but is uniquely equipped to do so as a Time Lady.
During one adventure, the TARDIS fell from N-Space into E-Space. While in E-Space, they happened upon Adric, a future companion, who made his way on board the TARDIS as a stowaway. It was once Adric was introduced that Romana showed more of her old personality, often trying to "civilize" Adric and chiding him for what she believed to be stupid questions.
Shortly after Adric arrived on the scene, Romana chose to stop traveling with the Doctor. While in E-Space, however, the trio and K9 encountered a few different threats: The Great Vampires, an old enemy of the Time Lords on a medieval-style planet, and later the Tharils, a once-powerful and domineering race capable of space and time travel without the need for a ship like the TARDIS who was now enslaved. At the end of this encounter, Romana II chose to leave traveling with The Doctor behind. She stayed in E-Space with K9 to help the Tharils rebuild their life in a more balanced way.
Romana's journey on the show ended in that episode, though she has been mentioned a few times since. In the expanded media universe of the show, Romana went on to become Lady President of the Time Lords and regenerated at least once more. How much of this is canon is, as always with Doctor Who, up for debate.
Although the Tenth Doctor mentioned that things went well for Romana until the Time War, there is always a possibility that she could return. Before the reveal of River Song's identity, a popular fan theory was that she was in fact a regenerated Romana. If she does return, she would still be than a match for The Doctor and could definitely hold her own as a companion or even a traveler on her own journey.