  In this paper, we study the rational cohomology rings of indefinite Kac-Moody
groups and their flag manifolds. By extracting the information of cohomology
from the Poincar\'{e} series, we are able to determine the rational cohomology
rings of Kac-Moody groups and their flag manifolds. Since Kac-Moody groups and
their flag manifolds are rational formal, we also determine their rational
homotopy groups and rational homotopy types.
