A multi-agent system (MAS) consists of multiple agents, each under the influence of a local rule that represents its interaction with other agents. Most recent research on discrete-time MASs concentrates on local rules that are linear and does not deal with communication constraints on the information exchange among agents. However, local interactions between agents in the real world are more likely governed by nonlinear rules and are in the presence of time-varying delays. This paper aims to investigate the consensus of a discrete-time MAS with transmission nonlinearity and time-varying delays. In particular, based on a representative general nonlinear model, we obtain several basic criteria for the consensus of the MAS. These results cover several existing results as their special cases. Moreover, the model we consider does not satisfy the convexity assumption which was commonly taken as an important condition for the consensus of discrete-time MASs. The assumptions we make on the nonlinear transmission function are necessary in the sense that, if they are not satisfied, a connected topology can be constructed that does not guarantee consensus. (C) 2013 Elsevier Ltd. All rights reserved.