  This paper focuses on the analysis of $4d$ $\mathcal{N}=4$ superconformal
theories in the presence of a defect from the point of view of the conformal
bootstrap. We will concentrate first on the case of codimension one, where the
defect is a boundary that preserves half of the supersymmetry. After studying
the constraints imposed by supersymmetry, we will obtain the Ward identities
associated to two-point functions of $\tfrac{1}{2}$-BPS operators and write
their solution as a superconformal block expansion. Due to a surprising
connection between spacetime and R-symmetry conformal blocks, our results not
only apply to $4d$ $\mathcal{N}=4$ superconformal theories with a boundary, but
also to three more systems that have the same symmetry algebra: $4d$
$\mathcal{N}=4$ superconformal theories with a line defect, $3d$
$\mathcal{N}=4$ superconformal theories with no defect, and $OSP(4^*|4)$
superconformal quantum mechanics. The superconformal algebra implies that all
these systems possess a closed subsector of operators in which the bootstrap
equations become polynomial constraints on the CFT data. We derive these
truncated equations and initiate the study of their solutions.
