  The primary purpose of this paper is to study the Wiener-type regularity
criteria for non-linear equations driven by integro-differential operators,
whose model is the fractional $p-$Laplace equation. In doing so, with the help
of tools from potential analysis, such as fractional relative Sobolev
capacities, Wiener type integrals, Wolff potentials, $(\alpha,p)-$barriers, and
$(\alpha,p)-$balayages, we first prove the characterizations of the fractional
thinness and the Perron boundary regularity. Then, we establish a Wiener test
and a generalized fractional Wiener criterion. Furthermore, we also prove the
continuity of the fractional superharmonic function, the fractional
resolutivity, a connection between $(\alpha,p)-$potentials and
$(\alpha,p)-$Perron solutions, and the existence of a capacitary function for
an arbitrary condenser.
