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It has been known for some time that the SL(2,R) WZWN model reduces toLiouville theory. Here we give a direct and physical derivation of this resultbased on the classical string equations of motion and the proper string size.This allows us to extract precisely the physical effects of the metric andantisymmetric tensor, respectively, on the {\it exact} string dynamics in theSL(2,R) background. The general solution to the proper string size is alsofound. We show that the antisymmetric tensor (corresponding to conformalinvariance) generally gives rise to repulsion, and it precisely cancels thedominant attractive term arising from the metric.  Both the sinh-Gordon and the cosh-Gordon sectors of the string dynamics innon-conformally invariant AdS spacetime reduce here to the Liouville equation(with different signs of the potential), while the original Liouville sectorreduces to the free wave equation. Only the very large classical string size isaffected by the torsion. Medium and small size string behaviours are unchanged.  We also find illustrative classes of string solutions in the SL(2,R)background: dynamical closed as well as stationary open spiralling strings, forwhich the effect of torsion is somewhat like the effect of rotation in themetric. Similarly, the string solutions in the 2+1 BH-AdS background withtorsion and angular momentum are fully analyzed.Comment: 24 pages including 4 postscript figures. Enlarged version including a  section on string solutions in 2+1 black hole background. To be published in  Phys. Rev. D., December 199

From the WZWN model to the Liouville equation: Exact string dynamics in a conformally invariant AdS background