We study 2D wedge wetting using a continuum interfacial Hamiltonian model w
hich is solved by transfer-matrix methods. For arbitrary binding potentials
,:we ale able to exactly calculate the wedge free energy and interface heig
ht distribution function and;:thus, can completely classify all types of cr
itical behavior. We show that critical filling is characterized by strongly
universal fluctuation dominated critical exponents, while complete filling
is determined by the geometry rather than fluctuation effects. Related phe
nomena for interface depinning from defect lines in the bulk are also consi
dered.