  We formulate the uniformisation problem underlying the geometry of
W_n-gravity using the differential equation approach to W-algebras. We
construct W_n-space (analogous to superspace in supersymmetry) as an (n-1)
dimensional complex manifold using isomonodromic deformations of linear
differential equations. The W_n-manifold is obtained by the quotient of a
Fuchsian subgroup of PSL(n,R) which acts properly discontinuously on a simply
connected domain in CP^{n-1}. The requirement that a deformation be
isomonodromic furnishes relations which enable one to convert non-linear
W-diffeomorphisms to (linear) diffeomorphisms on the W_n-manifold. We discuss
how the Teichmuller spaces introduced by Hitchin can then be interpreted as the
space of complex structures or the space of projective structures with real
holonomy on the W_n-manifold. The projective structures are characterised by
Halphen invariants which are appropriate generalisations of the Schwarzian.
This construction will work for all ``generic'' W-algebras.
