  This survey hinges on the interplay between regularity and approximation for
linear and quasi-linear fractional elliptic problems on Lipschitz domains. For
the linear Dirichlet integral Laplacian, after briefly recalling H\"older
regularity and applications, we discuss novel optimal shift theorems in Besov
spaces and their Sobolev counterparts. These results extend to problems with
finite horizon and are instrumental for the subsequent error analysis.
Moreover, we dwell on extensions of Besov regularity to the fractional
$p$-Laplacian, and review the regularity of fractional minimal graphs and
stickiness. We discretize these problems using continuous piecewise linear
finite elements and derive global and local error estimates for linear
problems, thereby improving some existing error estimates for both
quasi-uniform and graded meshes. We also present a BPX preconditioner which
turns out to be robust with respect to both the fractional order and the number
of levels. We conclude with the discretization of fractional quasi-linear
problems and their error analysis. We illustrate the theory with several
illuminating numerical experiments.
