Here's what we know:
* He is likely killed by Hamlet. Hamlet, in his rage and madness, kills Polonius (mistaking him for Claudius), and then later kills Rosencrantz and Guildenstern, who were sent by Claudius to spy on him.
* The play does not directly show the killings. The audience only hears about them later, through other characters.
Shakespeare leaves the exact details of their deaths open to interpretation. Some productions might depict their deaths on stage, while others might leave it to the audience's imagination.