  We consider interfacial phenomena accompanying bulk quantum phase transitions
in presence of surface fields. On general grounds we argue that the surface
contribution to the system free energy involves a line of singularities
characteristic of an interfacial phase transition, occurring below the bulk
transition temperature T_c down to T=0. This implies the occurrence of an
interfacial quantum critical regime extending into finite temperatures and
located within the portion of the phase diagram where the bulk is ordered. Even
in situations, where the bulk order sets in discontinuously at T=0, the
system's behavior at the boundary may be controlled by a divergent length scale
if the tricritical temperature is sufficiently low. Relying on an effective
interfacial model we compute the surface phase diagram in bulk spatial
dimensionality $d\geq 2$ and extract the values of the exponents describing the
interfacial singularities in $d\geq 3$.
