  We study the transport properties of Schr\"{o}dinger operators on
$\mathbb{R}^d$ and $\mathbb{Z}^d$ with potentials that are periodic in some
directions and compactly supported in the others. Such systems are known to
produce "surface states" that are confined near the support of the potential.
We show that, under very mild assumptions, a class of surface states exhibits
what we describe as directional ballistic transport, consisting of a strong
form of ballistic transport in the periodic directions and its absence in the
other directions. Furthermore, on $\mathbb{Z}^2$, we show that a dense set of
surface states exhibit directional ballistic transport. In appendices, we
generalize Simon's classic result on the absence of ballistic transport for
pure point states [36], and prove a folklore theorem on ballistic transport for
scattering states. In particular, this final result allows for a proof of
ballistic transport for a dense set subset of $\ell^2(\mathbb{Z}^2)$ for
periodic strip models.
