Matching is everywhere: students apply to universities, job seekers interview with companies, medical residents are paired with hospitals. In these two-sided markets, both sides have preferences, and we want a matching that is stable — no unmatched pair would secretly prefer each other over their assigned partners. This chapter will show you a simple and elegant algorithm that always finds such a match, and we’ll explore why it works, who benefits, and what happens when we add real-world twists like quotas, indifference, and even wages.