This work is concerned with stochastic consensus of multi-agent systems with additive measurement noises, where the information exchange among agents is described by a directed graph. By combining the algebraic graph theory, matrix theory and stochastic analysis, the stochastic weak and strong consensus are examined. For the case with general digraphs, we give the necessary and sufficient conditions for the almost sure strong consensus and show that the mean square strong consensus and almost sure strong consensus are equivalent. We also give some necessary conditions and sufficient conditions for the mean square and almost sure weak consensus. Especially, the necessary and sufficient conditions for the mean square weak consensus are given for the case with undirected graphs.