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From http://www.bbc.co.uk/doctorwho/classic/episodeguide:
The four time travellers are enjoying a rare holiday, staying at a villa not far from Rome in the year 64 AD. The Doctor soon becomes restless and sets off to visit the city, taking Vicki with him. In their absence, Ian and Barbara are kidnapped by slave traders.

Having been mistaken for the famous lyre player Maximus Pettulian and asked to perform at the Emperor Nero's Court, the Doctor has to devise ever more elaborate schemes to avoid revealing that he cannot actually play the instrument.

Ian meanwhile becomes a galley slave, while Barbara is sold to Nero's slave buyer Tavius at an auction in Rome. Ian and a fellow slave named Delos escape from the galley when it is wrecked in a storm and make their way to Rome to try to find and rescue Barbara.

There they are recaptured and forced to fight as gladiators in the arena. Events reach their climax when, by accidentally setting light to the Emperor's plans for the rebuilding of Rome, the Doctor gives him the idea of having the city razed to the ground. Nero plays the lyre while Rome burns, and the Doctor and Vicki and a reunited Ian and Barbara make their separate ways back to the villa.