  We present a coordinate-invariant approach, based on a Pauli-Villars measure,
to the definition of the path integral in two-dimensional conformal field
theory. We discuss some advantages of this approach compared to the operator
formalism and alternative path integral approaches. We show that our path
integral measure is invariant under conformal transformations and field
reparametrizations, in contrast to the measure used in the Fujikawa
calculation, and we show the agreement, despite different origins, of the
conformal anomaly in the two approaches. The natural energy-momentum in the
Pauli-Villars approach is a true coordinate-invariant tensor quantity, and we
discuss its nontrivial relationship to the corresponding non-tensor object
arising in the operator formalism, thus providing a novel explanation within a
path integral context for the anomalous Ward identities of the latter. We
provide a direct calculation of the nontrivial contact terms arising in
expectation values of certain energy-momentum products, and we use these to
perform a simple consistency check confirming the validity of the change of
variables formula for the path integral. Finally, we review the relationship
between the conformal anomaly and the energy-momentum two-point functions in
our formalism.
