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We study a renormalizable, general theory of dilatonic gravity (with akinetic-like term for the dilaton) interacting with scalar matter near twodimensions. The one-loop effective action and the beta functions for thisgeneral theory are written down. It is proven that the theory possesses anon-trivial ultraviolet fixed point which yields an asymptotically freegravitational coupling constant (at $\epsilon \rightarrow 0$) in this regime.Moreover, at the fixed point the theory can be cast under the form of astring-inspired model with free scalar matter. The renormalization of theJackiw-Teitelboim model and of lineal gravity in $2+\epsilon$ dimensions isalso discussed. We show that these two theories are distinguished at thequantum level. Finally, fermion-dilatonic gravity near two dimensions isconsidered.Comment: LaTeX, 13 pages, no figure

GENERAL DILATONIC GRAVITY WITH AN ASYMPTOTICALLY FREE GRAVITATIONAL COUPLING CONSTANT NEAR TWO DIMENSIONS