What does Odysseus make Penelope promise before he leaves in The Odyssey?
Before Odysseus embarks upon his journey to go fight in the Trojan War, he makes Penelope promise that she will re-marry and leave behind their home if he does not return by the time Telemachus (their son) has sprouted a beard (in other words, by the time he has reached puberty).
Despite making this promise, Penelope does not adhere to its terms; ironically, by breaking this promise, she is actually acting out of greater fidelity and integrity. When suitors swarm upon their home, desperate to marry Penelope, she cleverly puts them off for as long as possible and gives them an impossible task to distract them from her refusal to accept one of them. Penelope, in fact, remains faithful to her husband despite his tragically long absence; she waits twenty years for his return, and even then, puts Odysseus through a test to verify that he is truly her long-lost husband!
Odysseus made Penelope promise him that when their son Telemachus got the first sign of a beard she would marry, if her husband had not returned from war. When Telemachus sailed for Troy to look for proof if his father were dead or alive, he unfortunately returned to Ithaca with a beard on his face.
Look after my mother and father in the house as you do now...And when you see a beard on our boy's chin, marry whomsoever you fancy and leave your home. (Odysseus to Penelope, Homer, Odyssey 18.260).