  In this paper, for the development of the study of almost Gorenstein graded
rings, we discuss some relations between almost Gorensteinness of
Cohen--Macaulay homogeneous rings and their $h$-vectors. Concretely, for a
Cohen--Macaulay homogeneous ring $R$, we give a sufficient condition for $R$ to
be almost Gorenstein in terms of the $h$-vector of $R$ and we also characterize
almost Gorenstein homogeneous domains with small socle degrees in terms of the
$h$-vector of $R$. Moreover, we also provide the examples of almost Gorenstein
homogeneous domains arising from lattice polytopes.
