  We re-examine the use of sum rules in the extraction of light quark masses
and discuss a number of potential problems with existing analyses. The most
important issue is that of the overall normalization of the hadronic spectral
functions should not be fixed by assuming complete resonance dominance of the
continuum threshold region as it can overestimate the resonance contributions
to spectral integrals by factors as large as $\sim 5$. The second important
uncertainty comes from the assumed location, $s_0$, of the onset of duality
with perturbative QCD, as the extracted quark masses depend very sensitively on
this parameter. The assumption of duality and the requirement of positivity of
$\rho_5(s)$ imposes very severe constraints on the shape of the relevant
spectral function in the dual region, and leads to rigorous lower bounds for
$m_u+m_d$ as a function of $s_0$. In the extractions of $m_s$ we find that the
conventional choice of the value of $s_0$ is not physical. For a more
reasonable choice of $s_0$ we are not able to find a solution that is stable
with respect to variations of the Borel transform parameter. This problem can
be overcome only if the hadronic spectral function is determined up to
significantly larger values of $s$ than is currently possible. Finally, we also
estimate the error associated with the convergence of perturbative QCD
expressions used in the sum rule analyses. Our conclusion is that, taking all
of these issues into account, the resulting sum rule estimates for both
$m_u+m_d$ and $m_s$ could easily have uncertainties as large as a factor of 2.
