This paper studies the consensus problem for a class of general second-order multi-agent systems (MASs) with communication delay. We first consider the delay-free case and obtain a necessary and sufficient condition for consensus. Then, based on the obtained conditions for the delay-free case, we deduce an explicit formula for the delay margin of the consensus for the case with time delay using the relationship between the roots of the characteristic equation and the time delay parameter. In addition, we consider the special case where the second-order model is a double integrator. For this case, simpler consensus conditions under communication delay are provided. (C) 2016 Elsevier Ltd. All rights reserved.