  Interfacial fluctuation effects occuring at wedge and cone filling
transitions are investigated and shown to exhibit very different
characteristics. For both geometries we show how the conditions for observing
critical (continuous) filling are much less restrictive than for critical
wetting, which is known to require fine tuning of the Hamaker constants. Wedge
filling is critical if the wetting binding potential does not exhibit a local
maximum, whilst conic filling is critical if the integrated strength of the
potential is attractive. This latter scenario is particularly encouraging for
future experimental studies. Using mean-field and effective Hamiltonian
approaches, which allow for breather-mode fluctuations which translate the
interface up and down the sides of the confining geometry, we are able to
completely classify the possible critical behaviour (for purely thermal
disorder). For the three dimensional wedge, the interfacial fluctuations are
very strong and characterised by a universal roughness critical exponent
$\nu_{\perp} =1/4$ independent of the range of the forces. For the physical
dimensions d=2 and d=3, we show that the influence of the cone geometry on the
fluctuations at critical filling is to mimic the analogous interfacial
behaviour occuring at critical wetting in the strong-fluctuation regime. In
particular, for d=3 and for quite arbitary choices of intermolecular potential,
the filling height and roughness show the same critical properties as those
predicted for three dimensional critical wetting with short-ranged forces in
the large wetting parameter ($\omega>2$) regime.
