The edge event-triggered consensus control for multi-agent systems with single- or double-integrator dynamics under directed communication topologies is investigated in this paper. With the edge event-triggered sampling strategies, a class of novel consensus protocols is proposed to reduce control update frequency. The main features of the proposed edge event-triggered control include three aspects: (i) measurement errors are defined based on directed information flow and thus they allow that the communication topology is directed, (ii) each individual controller updates its output only when the associated edge event occurs and the control actions are independent of other agents, and (iii) the proposed edge event-triggered control does not exhibit Zeno behaviour and it is shown that the inter-event intervals are lower bounded by a positive constant. In the consensus analysis, matrix transformation techniques are used to convert the high-dimensional system into a low-dimensional system, and some sufficient consensus conditions are derived for the multi-agent systems with single- or double-integrator dynamics. Finally, the effectiveness of the edge event-triggered sampling strategies is illustrated by simulations.