In general, the Hurwitz numbers count the branched covers of the Riemann sphere with prescribed ramification data or, equivalently, the factorizations of a permutation with prescribed cycle structure data. In the present paper, the study of monotone orbifold Hurwitz numbers is initiated. These numbers are both variations of the orbifold case and generalizations of the monotone case. These two cases have previously been studied in the literature. We derive a cut-and-join recursion for monotone orbifold Hurwitz numbers, determine a quantum curve governing their wave function, and state an explicit conjecture relating them to topological recursion. Bibliography: 27 titles.