  We study the structure of a $3-$Leibniz algebra $T$ graded by an arbitrary
abelian group $G,$ which is considered of arbitrary dimension and over an
arbitrary base field $\bbbf.$ We show that $T$ is of the form
$T=\uu\oplus\sum_jI_j,$ with $\uu$ a linear subspace of $T_1,$ the homogeneous
component associated to the unit element $1$ in $G,$ and any $I_j$ a well
described graded ideal of $T,$ satisfying $$ [I_j, T, I_k] = [I_j, I_k, T] =
[T, I_j, I_k] = 0, $$ if $j\neq k.$ In the case of $T$ being of maximal length,
we characterize the gr-simplicity of the algebra in terms of connections in the
support of the grading.
