  The main aim of this paper is to develop general algebraic and cohomological
tools for the study of the local geometry of moduli and parameter spaces in
Algebraic Geometry, culminating in the so-called Hitchin (or KZ) (projective)
connection over the Moduli of curves and the proof of its flatness. Among other
things, we notably give a 'modular' (i.e. in terms of the modular problem)
construction of the differential operators- of any order- acting on
(general)modular vector bundles, and in particular for the Lie algebra of
vector fields on a (general) Moduli space, with its natural action on
functions. The methods involve mostly Lie algebras and their cohomology.
