  We use modern bootstrap techniques to study half-BPS line defects in 4d N=4
superconformal theories. Specifically, we consider the 1d CFT with OSP(4*|4)
superconformal symmetry living on such a defect. Our analysis is general and
based only on symmetries, it includes however important examples like Wilson
and 't Hooft lines in N=4 super Yang-Mills. We present several numerical bounds
on OPE coefficients and conformal dimensions. Of particular interest is a
numerical island obtained from a mixed correlator bootstrap that seems to imply
a unique solution to crossing. The island is obtained if some assumptions about
the spectrum are made, and is consistent with Wilson lines in planar N=4 super
Yang-Mills at strong coupling. We further analyze the vicinity of the
strong-coupling point by calculating perturbative corrections using analytic
methods. This perturbative solution has the sparsest spectrum and is expected
to saturate the numerical bounds, explaining some of the features of our
numerical results.
