  In general, the existence of a maximal green sequence is not mutation
invariant. In this paper we show that it is in fact mutation invariant for
cluster quivers of finite mutation type. In particular, we show that a mutation
finite cluster quiver has a maximal green sequence unless it arises from a
once-punctured closed marked surface, or one of the two quivers in the mutation
class of X7. We develop a procedure to explicitly find maximal green sequences
for cluster quivers associated to arbitrary triangulations of closed marked
surfaces with at least two punctures. As a corollary, it follows that any
triangulation of a marked surface with boundary has a maximal green sequence.
We also compute explicit maximal green sequences for exceptional quivers of
finite mutation type.
