  We investigate the box dimensions of the horizon of a fractal surface defined
by a function $f \in C[0,1]^2 $. In particular we show that a prevalent surface
satisfies the `horizon property', namely that the box dimension of the horizon
is one less than that of the surface. Since a prevalent surface has box
dimension 3, this does not give us any information about the horizon of
surfaces of dimension strictly less than 3. To examine this situation we
introduce spaces of functions with surfaces of upper box dimension at most
\alpha, for \alpha $\in$ [2,3). In this setting the behaviour of the horizon is
more subtle. We construct a prevalent subset of these spaces where the lower
box dimension of the horizon lies between the dimension of the surface minus
one and 2. We show that in the sense of prevalence these bounds are as tight as
possible if the spaces are defined purely in terms of dimension. However, if we
work in Lipschitz spaces, the horizon property does indeed hold for prevalent
functions. Along the way, we obtain a range of properties of box dimensions of
sums of functions.
