Zoe first appears in the serial The Wheel in Space, where she is the librarian on board Space Station W3, also known as the Wheel. When the Cybermen attack, she aids the Doctor and Jamie in defeating them before stowing away aboard the TARDIS. In David Whitaker's script for The Wheel in Space, Zoe's last name is spelled "Heriot", but the double-"r" misspelling is also seen in reference works.
Zoe's age is not given in the series, but according to initial publicity she was fifteen when she joined the TARDIS crew. She holds a degree in pure mathematics and is a genius, with intelligence scores comparable to the Doctor's. Coupled with her photographic memory and the advanced learning techniques of her era, this makes her somewhat like a human calculator, able to perform complicated mathematics in her head. Part of the reason for her wanting to travel with the Doctor is her chafing at the restrictions and sterile surroundings of her station-bound existence. However, her real world experience is severely limited, and that gives her an ability to frequently get herself in trouble.
Together with the Doctor and Jamie, she meets the Cybermen again when they invade 20th Century London, enters the surreal Land of Fiction, fights the Ice Warriors and survives the battlefields of the War Chief's war games. Her journeys with the Doctor come to an end in that serial, when the Time Lords finally catch up with the Doctor. As well as forcing a regeneration on him and exiling him to Earth, the Time Lords return Jamie and Zoe to their own times, wiping the memory of their experiences with the Doctor (save for their first encounters with him) in the process.