We'll break down the plots of the stories one-by-one, but first, a super-short, super-generalization of these stories: something goes wrong (or seems to go wrong) with a robot and three scientists—Susan Calvin and the Powell-Donovan team—figure out what the problem is and fix it (or figure out that it doesn't need fixing), and then everyone loves robots. However, there are at least two stories that don't at all fit in with that generalization: the frame story and "Robbie."
The frame story: Susan Calvin is being interviewed because she is retiring after 50 years at US Robots and Mechanical Men, Inc., and she tells the interviewer about what happened with robots over that time. (This frame comes back between the stories.)
"Robbie": In 1998, the robot Robbie is a nursemaid for a kid named Gloria, who loves Robbie a lot. Unfortunately, Gloria's mom doesn't love Robbie, partly because other people in the neighborhood don't like robots. Mom Grace convinces Dad George to get rid of Robbie, but that makes Gloria miserable. (He was, after all, the only member of the family whose name didn't start with a "G," which must have been a relief.) The parents take Gloria to see the robot factory to show her that robots aren't really people. Gloria nearly dies when she sees Robbie working there and tries to get to him—but Robbie saves her. This convinces everyone that Gloria should get to keep Robbie. (Although robots are then banned from Earth starting in 2003, as Calvin reminds us in the frame.)
"Runaround": In 2015, Gregory Powell and Michael Donovan are on Planet Mercury and they have a problem: they need some element to make sure their power generator works, but their state-of-the-art robot, Speedy, is running in circles and their older robots are useless. Why is Speedy acting weird? It's because the Second and Third Law are in conflict—he's got an order to do something, but he's also got to protect himself. The only way to snap him out of it is to put a human life in danger and so activate the First Law. Once the First Law is activated, it works to snap Speedy out of his confusion, and everyone loves robots.
"Reason": Half a year later, Powell and Donovan are on a space station seeing if the robot Cutie can be trusted to take over the management of the space station. This particular space station converts energy and beams it down to Earth and the colonies. The problem is that Cutie doesn't believe in Earth since he has no evidence of it, and only believes in what he can reason out. So Cutie starts worshipping the power converter—but his method of worshipping works out perfectly, and so everyone loves robots, even when they turn out to be religious fundamentalists.
"Catch that Rabbit": Another half a year later (it must be 2016 by now), Powell and Donovan are checking out a mining robot named Dave that controls other worker robots. The problem? Every time there's an emergency and people aren't around, Dave freaks out. Powell and Donovan cause a cave-in that almost kills them, but they figure out in time that Dave only melts down when he has to control all the other worker robots. So Powell and Donovan shoot one of the worker robots, which frees Dave up so that he can save them. And so everyone loves robots—including that robot that Powell and Donovan had to shoot.
"Liar!": In 2021, a robot named Herbie is created with the unintentional ability to read minds. He tells people what they want to hear: Susan Calvin learns that the cute guy likes her, while Peter Bogert, who wants a promotion, hears that he's going to get it. But then Calvin figures out that Herbie is only telling people what they want to hear. Why? Because Herbie can read minds and he knows that he'll hurt people if he tells the truth—and hurting people is against the First Law. Then Calvin points out to Herbie that he's in a situation where anything he does will break the First Law and Herbie goes insane. And so everyone loves robots? That doesn't seem to fit this story. Everyone hates Herbie—yeah, that's more like it.
"Little Lost Robot": In 2029 Susan Calvin and Peter Bogert are brought to a secret military base to find an unstable robot who a) has an altered version of the First Law imprinted in his positronic brain; b) had been ordered by an upset scientist to "go lose yourself"; and c) is hiding among nearly identical robots. They run a series of tests, but the robot outsmarts them all, until his robotic pride gets the better of him and they catch him. He tries to attack Susan Calvin, but the First Law still prohibits it. And so everyone loves robots—but maybe we shouldn't screw around with the First Law, guys.
"Escape!": Soon after "Little Lost Robot," Calvin helps the childish super-computer Brain to figure out how to build a hyperatomic drive (think: warp speed). Calvin tells Brain that people don't mind death, which is good for Brain, because the only way for a hyperatomic drive to work is for people to die temporarily. But it's also bad because finding that loophole in the First Law makes Brain go a little crazy—he becomes a practical joker. So Powell and Donovan come out to check the experimental ship, which works, although they don't like Brain's jokes so much. And so everyone loves robots—but everyone hates practical jokes.
"Evidence": In 2032 a political backroom dealer named Francis Quinn accuses Stephen Byerley of being a robot. Calvin points out that the only way for a human to prove that he's a human is to disobey the Three Laws. During a speech, Byerley is harassed by someone in the crowd, and Byerley hits him—which proves that he's human, because a robot cannot harm a human being. So Byerley wins the election. Except, as Calvin notes at the end, a robot can hurt another robot, so Byerley could still be a robot if the person he hit was actually just a robot. It's clear that Calvin thinks Byerley is a robot, but there's no evidence for the reader to judge. And so everyone loves Stephen Byerley, who may or may not be a robot.
"The Evitable Conflict": In 2052 Stephen Byerley, now World Co-Ordinator, talks to Calvin about some problems going on with the Machines, the super-computers that help to run the world economy. Calvin points out that the Machines do what they do in order to help people, and he shouldn't worry so much. And so everyone loves robots, including the Machines, who are pretty boring for robots.
Then, at the end, we get a tiny note from the interviewer: Susan Calvin, who saw robots grow from simple nursemaids to world-running super-computers, has died at the age of 82 in 2064. And we have no idea how she spent her six (or so) years of retirement.