Communication is supposed to improve multi-agent collaboration and overall performance in cooperative Multi-agent reinforcement learning (MARL). However, such improvements are prevalently limited in practice since most existing communication schemes ignore communication overheads (eg, communication delays). In this paper, we demonstrate that ignoring communication delays has detrimental effects on collaborations, especially in delay-sensitive tasks such as autonomous driving. To mitigate this impact, we design a delay-aware multi-agent communication model (DACOM) to adapt communication to delays. Specifically, DACOM introduces a component, TimeNet, that is responsible for adjusting the waiting time of an agent to receive messages from other agents such that the uncertainty associated with delay can be addressed. Our experiments reveal that DACOM has a non-negligible performance improvement over other mechanisms by making a better trade-off between the benefits of communication and the costs of waiting for messages.