  We consider the finite-size scaling of equilibrium droplet shapes for fluid
adsorption (at bulk two-phase co-existence) on heterogeneous substrates and
also in wedge geometries in which only a finite domain $\Lambda_{A}$ of the
substrate is completely wet. For three-dimensional systems with short-ranged
forces we use renormalization group ideas to establish that both the shape of
the droplet height and the height-height correlations can be understood from
the conformal invariance of an appropriate operator. This allows us to predict
the explicit scaling form of the droplet height for a number of different
domain shapes. For systems with long-ranged forces, conformal invariance is not
obeyed but the droplet shape is still shown to exhibit strong scaling
behaviour. We argue that droplet formation in heterogeneous wedge geometries
also shows a number of different scaling regimes depending on the range of the
forces. The conformal invariance of the wedge droplet shape for short-ranged
forces is shown explicitly.
