  We report the existence of zero energy surface states localized at zigzag
edges of $N$-layer graphene. Working within the tight-binding approximation,
and using the simplest nearest-neighbor model, we derive the analytic solution
for the wavefunctions of these peculiar surface states. It is shown that zero
energy edge states in multilayer graphene can be divided into three families:
(i) states living only on a single plane, equivalent to surface states in
monolayer graphene; (ii) states with finite amplitude over the two last, or the
two first layers of the stack, equivalent to surface states in bilayer
graphene; (iii) states with finite amplitude over three consecutive layers.
Multilayer graphene edge states are shown to be robust to the inclusion of the
next nearest-neighbor interlayer hopping. We generalize the edge state solution
to the case of graphite steps with zigzag edges, and show that edge states
measured through scanning tunneling microscopy and spectroscopy of graphite
steps belong to family (i) or (ii) mentioned above, depending on the way the
top layer is cut.
