This paper studies market clearing in matching markets. The model is non-cooperative, fully decentralized, and in Markov strategies. Workers and firms bargain with each other to determine who will be matched with whom and at what terms of trade. Once a worker-firm pair reach agreement, they exit the market. Alternative possible matches affect agents' bargaining positions. We ask under which conditions such markets clear efficiently and find that inefficiencies -- mismatch and delay -- feature frequently. Mismatch occurs whenever an agent's bargaining position is at risk of deteriorating. Delay occurs whenever agents expect their bargaining position to improve. Delay can be extensive and structured with vertically differentiated markets endogenously clearing from the top down.