Remarks on the physical states and the Chiral algebra of 2D gravity coupled to $C\leqslant 1$ matter

Some elaboration is given to the structure of physical states in 2D gravity coupled to $C\leqslant 1$ matter, and to the chiral algebra ($w_\infty$) of $C_M=1$ theory which has been found recently, in the continuum approach, by Witten and by Klebanov and Polyakov. It is shown then that the chiral algebra is being realized as well in the minimal models of gravity ($C_M<1$), so that it stands as a general symmetry of 2D gravity theories.