  Exclusive non-leptonic two-body decays of $B$ mesons have been studied
extensively in the past two decades within the framework of factorization.
However, the exploration of the corresponding three-body case has only started
recently, in part motivated by new data. We consider here the simplest
non-leptonic three-body $B$ decays from the point of view of factorization,
namely heavy-to-heavy transitions. We provide a careful derivation of the
SCET/QCDF factorized amplitudes to NNLO in $\alpha_s$, and discuss the
numerical impact of NLO and NNLO corrections. We then study the narrow-width
limit, showing that the three-body amplitude reproduces analytically the known
quasi-two-body decay amplitudes, and compute finite-width corrections. Finally,
we discuss certain observables that are sensitive to perturbative NLO and NNLO
corrections and to higher Gegenbauer moments of the dimeson LCDAs. This is the
first study of non-leptonic three-body $B$ decays to NNLO in QCD.
