Compared with traditional consensus,this paper studies the generalized consensus prob-lem for discrete-time multi-agent systems with directed topology and communication delay.Novel distributed consensus protocols with and without communication delay are designed.Based on the analysis of error dynamical system and graph theory,the generalized consensus is globally asymptot-ically achieved under suitable conditions without changing the zero row-sums property of Laplacian matrix in networks.Moreover,the sufficient conditions for generalized consensus of communication delay are obtained under directed connections.Finally,some simulations have been provided to verify the theoretical results.