In this important paper the authors nail down the last part of the boundary regularity for the Dirichlet problem of quasilinear elliptic equations. The problem has a long history starting with the fundamental work by N. Wiener in 1924. He solved completely the boundary regularity problem in the case of harmonic functions. The so-called Wiener integral criterion (the Wiener test) came out of these studies. V. Maz’ya [Vestn. Leningr. Univ. Math. 3, 225-242 (1976); transl. from Vestn. Leningr. Univ. 25, No. 13, 42-55 (1970; Zbl 0252.35024)] showed that a corresponding \(p\)-Wiener test gives a sufficient condition for the solvability of the Dirichlet problem associated with the quasilinear elliptic equation \(\text{div} A(t, \nabla u(x)) = 0\) where \(A(x,\xi) \approx | \xi |^ p\) for some \(p\), \(1 < p \leq n\), and \(n\) is the dimension of the underlying Euclidean space. In [P. Lindqvist and O. Martio, Acta Math. 155, 153-171 (1985; Zbl 0607.35042)] it was shown that the \(p\)-Wiener test gives a necessary condition for \(n - 1 < p \leq n\) and this paper provides this result for the full range of \(p\), \(1 < p \leq n\). A new and efficient tool in this connection is the Wolff potential [L. I. Hedberg and Th. Wolff, Ann. Inst. Fourier 33, 161-187 (1983; Zbl 0525.31005)]. As a byproduct the authors generalize many known results of thinness for ordinary superharmonic functions to the nonlinear situation.