  The surface states of Weyl semi-metals (SM's) consist of disjointed
Fermi-arcs. This unusual surface-Fermiology provides a fingerprint of the
topological features of the bulk Weyl-phase. Using a combination of
semiclassical and numerical methods, we show that, in contrast to naive
expectation, there are closed magnetic orbits involving the open- surface
Fermi-arcs. Below a critical field strength that depends on sample thickness,
these orbits produce periodic quantum oscillations of the density of states in
a magnetic field, enabling a variety of experimental probes of the
unconventional Fermi-arc surface states. The orbits are also essential for
reproducing the bulk chiral anomaly in a finite slab. These results are then
extended to the closely related and recently discovered 3D Dirac SM materials,
including Cd3As2 and Na3Bi, which are doubled copies of Weyl semi-metals
protected by crystal symmetry. Despite the fact that the protecting crystal
symmetry is broken by a surface, we show that Dirac materials can still host
unconventional surface-states, which can be detected in quantum oscillations
experiments.
