A lagrangian euclidean model of Drinfeld--Sokolov (DS) reduction leading to general W--algebras on a Riemann surface of any genus is presented. The background geometry is given by the DS principal bundle K associated to a complex Lie group G and an SL(2,\Bbb C) subgroup S. The basic fields are a hermitian fiber metric H of K and a (0,1) Koszul gauge field A^* of K valued in a certain negative graded subalgebra \goth x of \goth g related to \goth s. The action governing the H and A^* dynamics is the effective action of a DS field theory in the geometric background specified by H and A^*. Quantization of H and A^* implements on one hand the DS reduction and on the other defines a novel model of 2d gravity, DS gravity. The gauge fixing of the DS gauge symmetry yields an integration on a moduli space of DS gauge equivalence classes of A^* configurations, the DS moduli space. The model has a residual gauge symmetry associated to the DS gauge transformations leaving a given field A^* invariant. This is the DS counterpart of conformal symmetry. Conformal invariance and certain non perturbative features of the model are discussed in detail
