  Two-dimensional quantum gravity is identified as a second-class system which
we convert into a first-class system via the Batalin-Fradkin (BF) procedure.
Using the extended phase space method, we then formulate the theory in most
general class of gauges. The conformal gauge action suggested by David, Distler
and Kawai is derived from a first principle. We find a local, light-cone gauge
action whose Becchi-Rouet-Stora-Tyutin invariance implies Polyakov's curvature
equation $\partial_{-}R=\partial_{-}^{3}g_{++}=0$, revealing the origin of the
$SL(2,R)$ Kac-Moody symmetry. The BF degree of freedom turns out be dynamically
active as the Liouville mode in the conformal gauge, while in the light-cone
gauge the conformal degree of freedom plays that r{\^o}le. The inclusion of the
cosmological constant term in both gauges and the harmonic gauge-fixing are
also considered.
