  We discuss a covariant functional integral approach to the quantization of
the bosonic string. In contrast to approaches relying on non-covariant operator
regularizations, interesting operators here are true tensor objects with
classical transformation laws, even on target spaces where the theory has a
Weyl anomaly. Since no implicit non-covariant gauge choices are involved in the
definition of the operators, the anomaly is clearly separated from the issue of
operator renormalization and can be understood in isolation, instead of
infecting the latter as in other approaches. Our method is of wider
applicability to covariant theories that are not Weyl invariant, but where
covariant tensor operators are desired.
  After constructing covariantly regularized vertex operators, we define a
class of background-independent path integral measures suitable for string
quantization. We show how gauge invariance of the path integral implies the
usual physical state conditions in a very conceptually clean way. We then
discuss the construction of the BRST action from first principles, obtaining
some interesting caveats relating to its general covariance. In our approach,
the expected BRST related anomalies are encoded somewhat differently from other
approaches. We conclude with an unusual but amusing derivation of the value $D=
26$ of the critical dimension.
