  The QED contribution to the anomalous magnetic moments of electron and muon
are known very precisely up to the order $\alpha^4$. However, the knowledge of
$\alpha^5$ term will also be required when the precision of measurement
improves further. This paper reports the first systematic attempt to evaluate
the $\alpha^5$ term. Feynman diagrams contributing to this term can be
classified into six gauge-invariant sets which can be subdivided further into
32 gauge-invariant subsets. Thus far we have numerically evaluated all
integrals of 17 gauge-invariant subsets which contain light-by-light-scattering
subdiagrams and/or vacuum-polarization subdiagrams. They cover most of leading
terms of muon $g-2$ and lead to a preliminary result 663(20) $(\alpha/\pi)^5$,
which is 8.5 times more precise than the old estimate.
