  Conformal defects describe the universal behaviors of a conformal field
theory (CFT) in the presence of a boundary or more general impurities. The
coupled critical system is characterized by new conformal anomalies which are
analogous to, and generalize those of standalone CFTs. Here we study the
conformal $a$- and $c$-anomalies of four dimensional defects in CFTs of general
spacetime dimensions greater than four. We prove that under unitary defect
renormalization group (RG) flows, the defect $a$-anomaly must decrease, thus
establishing the defect $a$-theorem. For conformal defects preserving minimal
supersymmetry, the full defect symmetry contains a distinguished $U(1)_R$
subgroup. We derive the anomaly multiplet relations that express the defect
$a$- and $c$-anomalies in terms of the defect (mixed) 't Hooft anomalies for
this $U(1)_R$ symmetry. Once the $U(1)_R$ symmetry is identified using the
defect $a$-maximization principle which we prove, this enables a
non-perturbative pathway to the conformal anomalies of strongly coupled
defects. We illustrate our methods by discussing a number of examples including
boundaries in five dimensions and codimension-two defects in six dimensions. We
also comment on chiral algebra sectors of defect operator algebras and
potential conformal collider bounds on defect anomalies.
