  Starting from the covariant action for $W_3$ gravity, we discuss the BRST
quantization of $W_3$ gravity. Taking the chiral gauge the BRST charge has a
natural interpretation in terms of the quantum Drinfeld--Sokolov reduction for
$Sl(3,\re)$. Nilpotency of this charge leads to the KPZ formula for $W_3$. In
the conformal gauge, where the covariant action reduces to a Toda action, the
BRST charge is equivalent to the one recently constructed by Bershadsky et al.
