  In this thesis, we consider two approaches to the study of correlation
functions in one-dimensional defect Conformal Field Theories (dCFT$_1$), in
particular those defined by 1/2-BPS Wilson line defects in the three- and
four-dimensional superconformal theories relevant in the AdS/CFT
correspondence. In the first approach, we use the analytic conformal bootstrap
to evaluate two examples of defect correlators. The four-point correlator of
the displacement supermultiplet inserted on the 1/2-BPS Wilson line in ABJM
theory is computed to the third order in a strong-coupling expansion and
reproduces the explicit first-order Witten diagram calculations. The CFT$_1$
data are then extracted from this correlator, and the operator mixing is solved
at first order. Consequently, all-order results are derived for the part of the
correlator with the highest logarithm power, uniquely determining the
double-scaling limit. Then, the five-point correlator of $1/2$-BPS operators
inserted on the 1/2-BPS Wilson line in $\mathcal{N}=4$ super Yang-Mills are
studied. The superblocks are derived for all channels of the OPE, and the
five-point correlator is bootstrapped to first order in a strong coupling
expansion. The CFT$_1$ data are then extracted, confirming that operator mixing
does not affect the first-order anomalous dimension. The second approach
considers the general structure of correlators in effective theories in
AdS$_2$. All scalar $n$-point contact Witten diagrams for external operators of
integer conformal weight are computed. Effective theories in AdS$_2$ defined by
an interaction Lagrangian with an arbitrary number of derivatives are then
considered and solved to first order using a new formalism of Mellin amplitudes
for 1d CFTs. Finally, the cusped Wilson line discretised action is presented as
an alternative way to obtain non-perturbative data: through Lattice Field
Theory.
