The phase boundaries for corner wetting (filling) in square and diagonal la
ttice Ising models are exactly determined and show a universal shift relati
ve to wetting near the bulk criticality. More generally, scaling theory pre
dicts that the filling phase boundary shift for wedges and cones is determi
ned 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 h
igher dimensions using nonclassical local functional and mean-field theory.
Detailed numerical transfer matrix studies of the magnetization profile in
finite-size Ising squares support the conjectured connection between filli
ng and the strong-fluctuation regime of wetting.