Warning: Wikipedia contains spoilers
The story takes place on Trantor during the reign of Emperor Cleon I. It starts with Hari's presentation of a paper at a mathematics convention detailing how psychohistory might be possible. The Emperor of the Galactic Empire learns of it, and wants to use Hari for political gain. Hari fears this and goes on his "flight."
He is helped by reporter Chetter Hummin, who introduces him to Dors Venabili. Hari takes an immediate liking to Dors. During his flight, he would also meet a Raych (who he adopts as his son), and Yugo Amaryl (who would become Hari's partner in developing psychohistory).
The ending tells us that Hummin is actually Cleon's first minister Eto Demerzel, who we later learn is in fact R. Daneel Olivaw. Also, by the end of the novel, Hari also suspects Dors of being a robot too. (A theme that would later be picked up in Forward the Foundation)