Abstract：Gradient Einstein manifolds are complete manifolds with special metric structure. In this report we plan to consider four questions on gradient Einstein manifolds. 1) Estimates for various geometric quantities, in particular the estimates of the scalar curvature and the growth of the potential function play important roles in further study of gradient Einstein manifolds. 2) Spectral estimates on gradient Einstein solitons, the spectral gap and compact resolvent for the weighted Hodge Laplace operator on gradient Ricci solitons and quasi Einstein manifolds. 3) The geometric or topological properties at infinity for gradient Einstein manifolds. 4) Classification questions for gradient Einstein manifolds under suitable conditions.