  Conformal field theory finds applications across diverse fields, from
statistical systems at criticality to quantum gravity through the AdS/CFT
correspondence. These theories are subject to strong constraints, enabling a
systematic non-perturbative analysis. Conformal defects provide a controlled
means of breaking the symmetry, introducing new physical phenomena while
preserving crucial benefits of the underlying conformal symmetry. This thesis
investigates conformal line defects in both the weak- and strong-coupling
regimes. Two distinct classes of models are studied. First, we focus on the
supersymmetric Wilson line in $\mathcal{N}=4$ Super Yang--Mills, which serves
as an ideal testing ground for the development of innovative techniques such as
the analytic conformal bootstrap. The second class consists of magnetic lines
in Yukawa models, which have fascinating applications in $3d$ condensed-matter
systems. These systems have the potential to emulate phenomena observed in the
Standard Model in a low-energy setting.
