In the last years a considerable attention has been paid to the validity of classical Sobolev inequalities and embeddings when the underlying bounded domain \(D\) in \(\mathbb R^n\), \(n\geq 2\), need not have a smooth boundary \(\partial D\). Most of this work has been concerned with the situation in which the target space of the embedding is of Lebesgue or Hölder type. Much less is known about the situation in the limiting cases. Thus if \(\partial D\) is smooth it is well-known that the Sobolev space \(W^1_n(D)\) is embedded in the Orlicz space \(L_{\Phi}(D)\) with Young function \(\Phi\) with values which behave like \(\exp (t^{n/(n-1)})\) for large values of the argument \(t\). This embedding was also considered for Hölder spaces. In the paper under review spaces of functions that are larger than \(W^1_n(D)\), but are contained in \(\bigcap _{1n\) is contained in them. The main aim of the paper is to give conditions on \(D\) which are sufficient to ensure Sobolev inequalities yet allow the boundary of \(D\) to be quite rough. The first central result concerns a \(c_0\)-John domain (a domain \(D\) such that there exist \(x_0\in D\) and \(c_0\in (0,1]\) with the property that every \(x\in D\) can be joined to \(x_0\) by a rectifiable curve \(\gamma :[0,l]\to D\), parametrized by arc length, with \(d(\gamma (t), \partial D)\geq c_0t\) for all \(t\in [0,l]\)); the class of such domains is wide and includes Lipschitz domains and the Koch snowflake. The authors show that if \(D\) is a bounded \(c_0\)-John domain and \(u\) is a function on \(D\) such that \[ I(a,D):=\left ( \int _D |\nabla u(x)|^n\log ^{an}(e+|\nabla u(x)|) dx \right)^{1/n}0\), which does not depend on \(|\nabla u|\), such that \[ \int _D\exp \left [ A_1\exp \left ( A_2 \left ( \frac{|u(x)-u_D|}{I(D)} \right)^{n/(n-1)} \right) \right ]dx \leq A_3 \] where \[ I(D):=\left ( \int _D |\nabla u(x)|^n\log ^{n-1}(e+|\nabla u(x)|) dx \right)^{1/n}<\infty. \] This inequality seems to be new even in balls as far the authors (and the referee) know.