The Doctor and his companions are pursued through time and space by the Daleks on the desert planet Aridius and beyond.

The TARDIS materialises in Earth's future on a space beacon just before it is attacked by pirates. The travellers find themselves trapped in a sealed section of the beacon. It is blown apart and flown to where the pirates will plunder it of the precious mineral argonite. They witness a conflict between the pirates and the Interstellar Space Corps, led by General Hermack and Major Warne. The ISC are convinced that the pirates' mastermind is an innocent yet eccentric space mining pioneer named Milo Clancey, while their true leader is a man named Caven. Caven has a secret base on the planet Ta. He is assisted by Madeleine Issigri, daughter of Clancey's ex-partner Dom, who - unknown to her - is now his captive. When Madeleine discovers Caven's full treachery, she helps to bring him to justice. The time travellers are given a lift back to the TARDIS by Clancey in his rickety old ship, the LIZ 79.

On the planet Vortis, the Doctor and his companions aid the butterfly-like Menoptra in their battle against the parasitic Animus and its mind-controlled minions, the ant-like Zarbi.

When the TARDIS arrives on the plains of Asia Minor not far from the besieged city of Troy, the Doctor is hailed by Achilles as the mighty god Zeus and taken to the Greek camp. He meets Agamemnon and Odysseus. Forced to admit he is a mere mortal — albeit a traveller in space and time — he is given two days to devise a scheme to capture Troy. Steven and Vicki, meanwhile, have been taken prisoner by the Trojans. Vicki, believed to possess supernatural powers, is given two days to banish the Greeks to prove she is not a spy.

The travellers arrive in a strange domain presided over by the Celestial Toymaker — an enigmatic, immortal entity who forces them to play a series of games, failure at which will render them his playthings for all eternity.

Professor Monty Lafcado, haunted by the memory of his late mother, sets out to construct a mighty computer in his basement, much to the concern of his friend, Dr Pugh.