  Let $R=\bigoplus_{\alpha\in\Gamma}R_{\alpha}$ be a graded integral domain
graded by an arbitrary grading torsionless monoid $\Gamma$, and $\star$ be a
semistar operation on $R$. In this paper we define and study the graded
integral domain analogue of $\star$-Nagata and Kronecker function rings of $R$
with respect to $\star$. We say that $R$ is a graded Pr\"{u}fer
$\star$-multiplication domain if each nonzero finitely generated homogeneous
ideal of $R$ is $\star_f$-invertible. Using $\star$-Nagata and Kronecker
function rings, we give several different equivalent conditions for $R$ to be a
graded Pr\"{u}fer $\star$-multiplication domain. In particular we give new
characterizations for a graded integral domain, to be a P$v$MD.
