Conformal defects break part of the symmetry of a bulk CFT. The broken
Ward identities lead to very general sum rules on the defect CFT data as
well as on the data of bulk operators in the presence of a defect. We
call these sum rules "defect soft theorems", and they hold generally for
defects which break conformal symmetry, flavor symmetry, or
supersymmetry. In this talk I will focus on line defects for which we
can rewrite the constraints in dispersive sum rule form, and show how
the defect soft theorems impose constraints on the defect spectrum and
OPE coefficients. I will also show how to combine them with the
numerical conformal bootstrap for defects, a nonperturbative and
rigorous method to obtain CFT data. I will highlight two examples: 1) a
flux tube in pure YM in AdS_3, where the bootstrap can be used to test
the validity of a smooth interpolation between perturbative models at
small and large AdS radius. 2) the ½-BPS Maldacena-Wilson line in N=4
SYM, where an interplay with integrability and supersymmetric
localization provides a unique opportunity to obtain exact results.