  We study scaling and renormalization in two dimensional quantum gravity in a
covariant framework. After reviewing the definition of a proper path integral
measure, we use scaling arguments to rederive the KPZ relations, the fractal
dimension of the theory and the scaling of the reparametrization-invariant two
point function. Then we compute the scaling exponents entering in this
relations by means of the functional RG. We show that a key ingredient to
obtain the correct results already known from Liouville theory is the use of
the exponential parametrization for metric fluctuations. We also show that with
this parametrization we can recover the correct finite part of the effective
action as the $\epsilon \to 0$ continuation of gravity in $d=2+\epsilon$
dimensions.
