  The ${\rm SL}_3$-skein algebra $\mathscr{S}_{\bar{q}}(\mathfrak{S})$ of a
punctured oriented surface $\mathfrak{S}$ is a quantum deformation of the
coordinate algebra of the ${\rm SL}_3$-character variety of $\mathfrak{S}$.
When $\bar{q}$ is a root of unity, we prove the Unicity Theorem for
representations of $\mathscr{S}_{\bar{q}}(\mathfrak{S})$, in particular the
existence and uniqueness of a generic irreducible representation. Furthermore,
we show that the center of $\mathscr{S}_{\bar{q}}(\frak{S})$ is generated by
the peripheral skeins around punctures and the central elements contained in
the image of the Frobenius homomorphism for $\mathscr{S}_{\bar{q}}(\frak{S})$,
a surface generalization of Frobenius homomorphisms of quantum groups related
to ${\rm SL}_3$. We compute the rank of $\mathscr{S}_{\bar{q}}(\mathfrak{S})$
over its center, hence the dimension of the generic irreducible representation.
