  The growth of stochastic interfaces in the vicinity of a boundary and the
non-trivial crossover towards the behaviour deep in the bulk is analysed. The
causal interactions of the interface with the boundary lead to a roughness
larger near to the boundary than deep in the bulk. This is exemplified in the
semi-infinite Edwards-Wilkinson model in one dimension, both from its exact
solution and numerical simulations, as well as from simulations on the
semi-infinite one-dimensional Kardar-Parisi-Zhang model. The non-stationary
scaling of interface heights and widths is analyzed and a universal scaling
form for the local height profile is proposed.
