Set in the Greek kingdom of Epirus, Andromache presents the complex interactions of Greeks and Trojans who are affected by the Trojan War, which ended one year prior to the play's opening. Fearing future reprisals, the Greeks are concerned that Astyanax, the son of Andromache and her deceased husband, Hector (the famed hero of Troy), will grow up to seek vengeance for his father’s death and the fall of Troy. They send Oreste to Epirus to kill Astyanax. Andromache and Astyanax are living there, enslaved by Pyrrhus, the king of Epirus and the son of the Greek warrior Achilles. Pyrrhus is engaged to Hermione, whose parents were Helen of Troy and Menelaus. Oreste, however, also loves Hermione, while Pyrrhus is besotted with Andromache.

Pyrrhus first argues against the killing of an innocent child—then he threatens Andromache with doing so unless she agrees to marry him. Hermione, disturbed by her fiancé’s attraction to the other woman, wants Andromache killed along with her Astyanax. Planning to enact her own vengeance if necessary, Hermione makes Oreste promise to kill Pyrrhus if Pyrrhus throws her over for Andromache. Unrest begins to stir among the people of Epirus, who resent the elite’s absorption in their personal lives at the expense of the general population's well-being.

Andromache is tortured by nightmares of her dead husband and worries over losing her son. She decides that she will break her vow of continued fidelity to the late Hector and marry Pyrrhus in order to save Astyanax’s life. After the wedding, she will kill herself. Hermione promises herself to Oreste in exchange for killing Pyrrhus. Andromache is saved from her fate when Oreste has his men kill Pyrrhus. Hermione, however, has changed her mind and decided she loves Pyrrhus. She rejects Oreste, who, before escaping from Epirus, loses his mind. Hermione, overcome with remorse for her actions, takes her own life. Andromache will now reign as Queen of Epirus.