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We classify n -hereditary monomial algebras in three natural contexts: First, we give a classification of the n -hereditary truncated path algebras. We show that they are exactly the n -representation-finite Nakayama algebras classified by Vaso. Next, we classify partially the n -hereditary quadratic monomial algebras. In the case n = 2 , we prove that there are only two examples, provided that the preprojective algebra is a planar quiver with potential. The first one is a Nakayama algebra and the second one is obtained by mutating k A 3 ⊗ k k A 3 , where A 3 is the Dynkin quiver of type A with bipartite orientation. In the case n ≥ 3 , we show that the only n -representation finite algebras are the n -representation-finite Nakayama algebras with quadratic relations. [ABSTRACT FROM AUTHOR]