  We examine the QCD corrections to the structure functions $F_2^\gamma(x,Q^2)$
and $F_L^\gamma(x,Q^2)$ for a real photon target. The pointlike photon
contributions from light and heavy quarks are computed through
$O(\alpha\alpha_S)$. A parameterization of the hadronic (resolved) photon
contribution is also included and a comparison is made with the present
experimental data. We find that while the pointlike contributions are large for
$F_2^\gamma(x,Q^2)$, they are the dominant part of $F_L^\gamma(x,Q^2)$. For
$Q^2 > 50$ $({\rm GeV}/c)^2$ the charm component is as least as large as the
light quark component.
