How It All Goes Down
The play opens at the offices of Rossum's Universal Robots. Helena Glory, daughter of the president, comes to visit the factory. She is greeted by Harry Domin, who agrees to show her around. Domin talks a lot (though, to be fair, that's what you do in a play) and tells her that the factory is producing lots of robots, who will change the world and make labor so cheap that all work and poverty will be eliminated. (Hmm, we wonder if that's going to turn out the way he wants…)
Helena wants to convince the robots to revolt… but the robots are completely uninterested. Instead, Harry Domin falls in love with her and convinces her to marry him. (Don't do it Helena! He's a jerk!)
Too late—flash forward ten years; Helena is still at the factory with her husband, Domin. Things have gone horribly awry, though. In the first place, people have stopped having children, which various characters connect to the fact that robots do everything now, so humans don't need to reproduce. Helena is upset about humans having no more babies, so she burns the formula for making robots. There are no copies of this formula because… well, otherwise the plot wouldn't work. (Čapek is maybe not the greatest plotter in the history of theater.)
Other problems ensue: The robots have decided that they're sick and tired of being ordered about, and so decide to kill all humans on earth. You'd think maybe there could be some middle ground between slavery and murdering everyone on the planet, but this isn't a middle-ground kind of play.
The humans, led by Domin, first plan to escape by boat, but they're surrounded. Then they figure that they'll trade the formula for creating robots for their safety. But Helena's burned the formula—oops. So, yeah, the robots kill everyone.
Everyone except… Alquist, that is—he's a chief of construction at R.U.R. He wanted to die, but instead the robots set him up to try to find the formula for making new robots. Without the formula, the robots can't build more robots, which means when the old robots wear out (in 20 years or so), that's it for life (or their version of life) on earth. Alquist tries to re-create the formula, but he can't (maybe because he's spending so much time in over-heated soliloquies). He also tells the robots to bring him humans, but they can't, because all the humans are dead.
But, wait! Two robots—Primus and Helena—seem to have souls (perhaps given to them by the physiologist Dr. Gall before he was killed). Alquist discovers that these robots are in love and decides that this means that the world is saved. He sends them off to produce life, which doesn't actually make much sense. Just because robots have fallen in love doesn't mean they can make babies. (Alquist isn't a biologist, though, so maybe he's unclear on the details.) In any event, he's happy because he thinks love and life will go on. The end.