This is a joint work with Tian. We study the structure of the limit space of a sequence of almost
Einstein manifolds, which are generalizations of Einstein manifolds. Roughly speaking, such manifolds are the
initial manifolds of some normalized Ricci flows whose scalar curvatures are almost constants over space-time in the
$L^1$-sense, Ricci curvatures are bounded from below at the initial time. Under the non-collapsed condition, we show that the limit space of a
sequence of almost Einstein manifolds has most properties which is known for the limit space of Einstein
manifolds. As applications, we can apply our structure results to study the
properties of K\"ahler manifolds.