  Let $R$ be a graded commutative ring with non-zero unity $1$ and $M$ be a
graded unitary $R$-module. Let $GS(M)$ be the set of all graded $R$-submodules
of $M$ and $\phi: GS(M)\rightarrow GS(M)\bigcup\{\emptyset\}$ be a function. A
proper graded $R$-submodule $K$ of $M$ is said to be a graded $\phi-$prime
$R$-submodule of $M$ if whenever $r$ is a homogeneous element of $R$ and $m$ is
a homogeneous element of $M$ such that $rm\in K-\phi(K)$, then either $m\in K$
or $r\in (K:_{R}M)$. If $\phi(K)=\emptyset$ for all $K\in GS(M)$, then a graded
$\phi-$prime submodule is exactly a graded prime submodule. If $\phi(K)=\{0\}$
for all $K\in GS(M)$, then a graded $\phi-$prime submodule is exactly a graded
weakly prime submodule. Several properties of graded $\phi-$prime submodules
have been investigated.
