Deadlines and fixed end dates are pervasive in matching markets including school choice, the market for new graduates, and even financial markets such as the market for federal funds. Deadlines drive fundamental non-stationarity and complexity in behavior, generating significant departures from the steady-state equilibria usually studied in the search and matching literature. I consider a two-sided matching market with search frictions where vertically differentiated agents attempt to form bilateral matches before a deadline. I give conditions for existence and uniqueness of equilibria, and show that all equilibria exhibit an ``anticipation effect'' where less attractive agents become increasingly choosy over time, preferring to wait for the opportunity to match with attractive agents who, in turn, become less selective as the deadline approaches. When payoffs accrue after the deadline, or agents do not discount, a sharp characterization is available: at any point in time, the market is segmented into a first class of matching agents and a second class of waiting agents. This points to a different interpretation of unraveling observed in some markets and provides a benchmark for other studies of non-stationary matching. A simple intervention -- a small participation cost -- can dramatically improve efficiency.