  This paper introduces a new type of open book decomposition for a contact
three-manifold with a specified characteristic foliation $\mathcal{F}_\xi$ on
its boundary. These \textit{foliated open books} offer a finer tool for
studying contact manifolds with convex boundary than existing models, as the
boundary foliation carries more data than the dividing set. In addition to
establishing fundamental results about the uniqueness and existence of foliated
open books, we carefully examine their relationship with the partial open books
introduced by Honda-Kazez-Matic. Foliated open books have user-friendly cutting
and gluing properties, and they arise naturally as submanifolds of classical
open books for closed three-manifolds. We define three versions of foliated
open books (embedded, Morse, and abstract), and we prove the equivalence of
these models as well as a Giroux Correspondence which characterizes the
foliated open books associated to a fixed triple $(M, \xi, \mathcal{F})$.
