  We study three-point correlation functions of scalar operators in conformal
field theories with boundaries and interfaces. We focus on two cases where
there are one bulk and two boundary operators (B$\partial\partial$), or two
bulk and one boundary operators (BB$\partial$). We perform a detailed analysis
of the conformal blocks in different OPE channels. In particular, we obtain the
bulk channel conformal blocks of the BB$\partial$ three-point functions for
arbitrary exchanged spins in a series expansion with respect to the radial
coordinates. We also study examples of such three-point functions in the
simplest holographic dual where the $AdS_{d+1}$ space contains a brane filling
an $AdS_d$ subspace. Such a setup arises in top-down models with probe branes
and is also relevant for the functional approach to boundary and interface CFT
correlators. We systematically study the Witten diagrams in this setup both in
position space and in Mellin space. We also discuss in detail how to decompose
these Witten diagrams into conformal blocks.
