Astronauts rescued from a mission to mars are kidnapped when they arrive back on earth. They seem to have been irradiated, and can now kill people merely by touching them. But the Doctor discovers the astronauts have been replaced with Alien ambassadors who are being forced to behave in a hostile manner. With The Doctor traveling to the alien mother ship, and Liz kidnapped, The Brigadier is left wondering who to trust.

The TARDIS lands in a space museum on Earth in the late 21st century, where the Second Doctor, Jamie and Zoe learn that contact has been lost between Earth and the Moon. In this era, instant travel — T-Mat — has revolutionised the Earth. Its people have lost interest in space travel. The Doctor and his companions travel to the Moon in an old-style rocket and reach the Moonbase, control centre for T-Mat, only to find a squad of Ice Warriors have commandeered the base and plan to use the T-Mat network to their advantage.

The TARDIS arrives in London in 1966 and the First Doctor and Dodo visit the Post Office Tower. There they meet Professor Brett, whose revolutionary new computer WOTAN (Will Operating Thought ANalogue) can actually think for itself and is shortly to be linked up to other major computers around the world — a project overseen by civil servant Sir Charles Summer.