Medea is centered on a wife’s calculated desire for revenge against her unfaithful husband. The play is set in Corinth some time after Jason's quest for the Golden Fleece, where he met Medea. The play begins with Medea raging at Jason for arranging to marry Glauce, the daughter of Creon (king of Corinth). The nurse, overhearing Medea’s grief, fears what she might do to herself or her children.
Creon, in anticipation of Medea’s wrath, arrives and reveals his plans to send her into exile. Medea pleads for one day’s delay and eventually Creon acquiesces. In the next scene Jason arrives to explain his rationale for his apparent betrayal. He explains that he couldn’t pass up the opportunity to marry a royal princess, as Medea is only a barbarian woman, but hopes to someday join the two families and keep Medea as his mistress. Medea, and the chorus of Corinthian women, do not believe him. She reminds him that she left her own people for him ("I am the mother of your children. Whither can I fly, since all Greece hates the barbarian?"), and that she saved him and slew the dragon. Jason promises to support her after his new marriage, but Medea spurns him: "Marry the maid if thou wilt; perchance full soon thou mayst rue thy nuptials."
In the following scene Medea encounters Aegeus, King of Athens. He reveals to her that despite his marriage to his wife he is still without children. He visited the oracle who merely told him that he was instructed “not to unstop the wineskin’s neck.” Medea relays her current situation to him and begs for Aegeus to let her stay in Athens if she gives him drugs to end his infertility. Aegeus, unaware of Medea’s plans for revenge, agrees.
Medea then returns to plotting the murders of Glauce and Creon. She decides to poison some golden robes (a family heirloom and gift from the sun god Helios) and a coronet, in hopes that the bride will not be able to resist wearing them, and consequently be poisoned. Medea resolves to kill her own children as well, not because the children have done anything wrong, but because she feels it is the best way to hurt Jason. She calls for Jason once more and, in an elaborate ruse, apologizes to him for overreacting to his decision to marry Glauce. When Jason appears fully convinced that she regrets her actions, Medea begins to cry in mourning of her exile. She convinces Jason to allow her to give the robes to Glauce in hopes that Glauce might get Creon to lift the exile. Eventually Jason agrees and allows their children to deliver the poisoned robes as the gift-bearers.
Forgive what I said in anger! I will yield to the decree, and only beg one favor, that my children may stay. They shall take to the princess a costly robe and a golden crown, and pray for her protection.
In the next scene a messenger recounts Glauce and Creon’s deaths. When the children arrived with the robes and coronet Glauce put them on gleefully and went to find her father. Soon the poisons overtook Glauce and she fell to the floor, quickly dying. Creon clutched her tightly as he tried to save her and, by coming in contact with the robes and coronet, got poisoned and died as well.
Alas! The bride had died in horrible agony; for no sooner had she put on Medea's gifts than a devouring poison consumed her limbs as with fire, and in his endeavor to save his daughter the old father died too.
While Medea is content with her current success she decides to take it one step forward. Since Jason brought shame upon her for trying to start a new family, Medea resolves to destroy the family he was willing to give up by killing their sons. Medea does have a moment of hesitation when she considers the pain that her children’s deaths will put her through. However, she steels her resolve to cause Jason the most pain possible and rushes offstage with a knife to kill her children. As the chorus laments her decision, the children are heard screaming. Jason then rushes onto the scene to confront Medea about murdering Creon and Glauce and he quickly discovers that his children have been killed as well. Medea then appears above the stage with the bodies of her children in the chariot of the sun god Helios. When this play was put on, this scene was accomplished using the mechane device usually reserved for the appearance of a god or goddess. She confronts Jason, reveling in his pain at being unable to ever hold his children again:
"I do not leave my children's bodies with thee; I take them with me that I may bury them in Hera's precinct. And for thee, who didst me all that evil, I prophesy an evil doom."
She escapes to Athens with the bodies. The chorus is left contemplating the will of Zeus in Medea's actions:
Manifold are thy shapings, Providence!
Many a hopeless matter gods arrange.
What we expected never came to pass,
What we did not expect the gods brought to bear;
So have things gone, this whole experience through!

Question: Why does Medea resolve to kill her children?
Answer: Because it will hurt Jason.