  In this paper, we present a new formulation of topological conformal gravity
in four dimensions. Such a theory was first considered by Witten as a possible
gravitational counterpart of topological Yang-Mills theory, but several
problems left it incomplete. The key in our approach is to realise a theory
which describes deformations of conformally self-dual gravitational instantons.
We first identify the appropriate elliptic complex which does precisely this.
By applying the Atiyah-Singer index theorem, we calculate the number of
independent deformations of a given gravitational instanton which preserve its
self-duality. We then quantise topological conformal gravity by BRST
gauge-fixing, and discover how the quantum theory is naturally described by the
above complex. Indeed, it is a process which closely parallels that of the
Yang-Mills theory, and we show how the partition function generates an uncanny
gravitational analogue of the first Donaldson invariant.
