  We investigate critical wetting transitions for fluids adsorbed in wedge-like
geometries where the substrate height varies as a power-law, $z(x,y) \sim |x|
^\gamma$, in one direction. As $\gamma$ is increased from 0 to 1, the substrate
shape is smoothly changed from a planar-wall to a linear wedge. The continuous
wetting and filling transitions pertinent to these limiting geometries are
known to have distinct phase boundaries and critical singularities. We predict
that the intermediate critical wetting behaviour occurring for $0<\gamma< 1$
falls into one of {\it{three}} possible regimes depending on the values of
$\gamma$, p and q. The unbinding behaviour is characterised by a high degree of
non-universality, strongly anisotropic correlations and enhanced interfacial
roughness. The shift in phase boundary and emergence of universal critical
behaviour in the linear wedge limit is discussed in detail.
