  The phase boundaries for corner wetting (filling) in square and diagonal
lattice Ising models are exactly determined and show a universal shift relative
to wetting near the bulk criticality. More generally, scaling theory predicts
that the filling phase boundary shift for wedges and cones is determined by a
universal scaling function $R_d(\psi)$ depending only on the opening angle
$2\psi$. $R_d(\psi)$ is determined exactly in $d=2$ and approximately in higher
dimensions using non-classical local functional and mean-field theory. Detailed
numerical transfer matrix studies of the the magnetisation profile in
finite-size Ising squares support the conjectured connection between filling
and the strong-fluctuation regime of wetting.
