2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/53456These lecture notes review current progress on the class of conformal theories which may be studied by quantizing the conformal Toda dynamics. After summarizing recent developments in undertanding the quantum group structure of the Liouville theory, one recalls how classically, two-dimensional black holes come out from the non-abelian Toda systems, and reviews the geometrical interpretation of the classical $A_n$-Toda theories, just put forward, that relates W-geometries with the external geometries of holomorphic surfaces in complex projective spaces.22 pages, no figures. Notes of lecture at the September Rome meeting String Theory, Quantum Gravity, and the Unification of Fundamental InteractionsHigh Energy Physics - Theorytext