  In this paper we propose a unified approach, based on limiting interpolation,
to investigate the embeddings for the Sobolev space
$(\dot{W}^k_p(\mathcal{X}))_0, \, \mathcal{X} \in \{\mathbb{R}^d, \mathbb{T}^d,
\Omega\}$, in the subcritical case ($k < d/p$), critical case ($k = d/p$) and
supercritical case ($k > d/p$). We characterize the Sobolev embeddings in terms
of pointwise inequalities involving rearrangements and moduli of
smoothness/derivatives of functions and via extrapolation theorems for
corresponding smooth function spaces. Applications include Ulyanov-Kolyada type
inequalities for rearrangements, inequalities for moduli of smoothness, sharp
Jawerth-Franke embeddings for Lorentz-Sobolev spaces, various characterizations
of Gagliardo-Nirenberg, Trudinger, Maz'ya-Hansson-Brezis-Wainger and
Brezis-Wainger embeddings, among others. In particular, we show that the Tao's
extrapolation theorem holds true in the setting of Sobolev inequalities. This
gives a positive answer to a question recently posed by Astashkin and Milman.
