  In this paper we introduce a mass for asymptotically flat manifolds by using
the Gauss-Bonnet curvature. We first prove that the mass is well-defined and is
a geometric invariant, if the Gauss-Bonnet curvature is integrable and the
decay order $\tau$ satisfies $\tau > \frac {n-4}{3}.$ Then we show a positive
mass theorem for asymptotically flat graphs over ${\mathbb R}^n$. Moreover we
obtain also Penrose type inequalities in this case.
