Rudy is the son of Leopold and Molly Bloom. He died very shortly after birth. Ever since his death, roughly ten years ago, Leopold has been unable to enjoy sex with Molly. In "Hades," "Sirens," and "Oxen of the Sun," Bloom thinks of his dead son and tries to imagine what he might have been like if he were alive.
At the close of the "Circe" episode, we see Bloom imagine Rudy as an Eton schoolboy. Bloom thinks he sees his son in the street and calls out to him, but Rudy does not respond. Throughout the novel, it is clear that the lack of a son contributes to Bloom's feelings of impotence and unimportance.