We study non-stationary dynamic decentralized markets with adverse selection in which trade is bilateral and prices are determined by bargaining. Examples include labor markets, housing markets, and markets for financial assets. We characterize equilibrium, and identify the dynamics of transaction prices, trading patterns, and the average quality in the market. When the horizon is finite, the surplus in the unique equilibrium exceeds the competitive surplus; as traders become perfectly patient the market becomes completely illiquid at all but the first and last dates, but the surplus remains above the competitive surplus. When the horizon is infinite, the surplus realized equals the static competitive surplus. We study policies aimed at improving market performance, and show that subsidies to low quality or to trades at a low price, taxes on high quality, restrictions on trading opportunities, or government purchases may raise the surplus. In contrast, interventions like the Public-Private Investment Program for Legacy Assets reduce the surplus when traders are patient.