We construct a gauge theory of w gravity coupled to scalars. For scalar fields valued in some-finite-dimensional Lie algebra, a complete realisation of gauged w can be given using a number of gauge fields equal to the rank of the Lie algebra. We show how the theory can be truncated to describe a gauge theory of WN gravity, and we give the explicit result for W3. We show that the transformations of the scalar fields can be interpreted as a non-linear realisation on the coset w1+ w. 1990.
