  For nonlinear operators of fractional $p$-Laplace type, we consider two types
of solutions to the nonlocal Dirichlet problem: Sobolev solutions based on
fractional Sobolev spaces and Perron solutions based on superharmonic
functions. These solutions give rise to two different concepts of regularity
for boundary points, namely Sobolev and Perron regularity. We show that these
two notions are equivalent and we also provide several characterizations of
regular boundary points. Along the way, we give a new definition of Perron
solutions, which is applicable to arbitrary exterior Dirichlet data $g:
\Omega^c \to [-\infty,\infty]$. We obtain resolutivity and invariance results
for these Perron solutions, and show that the Sobolev and Perron solutions
coincide for a large class of exterior Dirichlet data. A uniqueness result for
the Dirichlet problem is also obtained for the class of bounded solutions
taking prescribed continuous exterior data quasieverywhere on the boundary.
