Duke Vincentio is frustrated that his people don't obey the law. Instead of cracking down on crime himself, he appoints a strict deputy, Angelo, to clean up Vienna.
Angelo doesn't waste any time. He busts Angelo for "fornicating" (having sex outside of marriage) and sentences him to death. When Claudio's sister, Isabella, tries to intervene, Angelo propositions her, offering to let Claudio off the hook if she agrees to hook up with Angelo.
The Duke steps in and confronts Angelo, who repents. Claudio is saved, and the Duke orders three couples to get married (solving the whole "sex outside of marriage" problem).