  Classical Hurwitz numbers count branched covers of the Riemann sphere with
prescribed ramification data, or equivalently, factorisations in the symmetric
group with prescribed cycle structure data. Monotone Hurwitz numbers restrict
the enumeration by imposing a further monotonicity condition on such
factorisations. In this paper, we prove that monotone Hurwitz numbers arise
from the topological recursion of Eynard and Orantin applied to a particular
spectral curve. We furthermore derive a quantum curve for monotone Hurwitz
numbers. These results extend the collection of enumerative problems known to
be governed by the paradigm of topological recursion and quantum curves, as
well as the list of analogues between monotone Hurwitz numbers and their
classical counterparts.
