For 1≤p≤∞ and n/p<α<1 + n/p, the fractional capacity Cα,p(K,Ω) of a compact subset K with respect to a bounded open Lipschitz set Ω⊃K of \(\mathbb R^{n}\) is systematically investigated and effectively applied to the fractional Sobolev space Wα,p(Ω) which plays an important role in calculus of variations, harmonic analysis, potential theory, partial differential equations, mathematical physics and so forth. In particular, we explore: the basic properties of Cα,p(K,Ω); the existence of a capacitary potential, and the relationship between Cα,p(K,Ω) and the fractional Laplace equation. the Cα,p(K,Ω)-based nature of a nonnegative Radon measure μ given on Ω that induces an embedding of Wα,p(Ω) into the Lebesgue space Lq(Ω,μ) as 1≤p≤q<∞ or the exponentially integrable Lebesgue space \(\exp (c|f|^{\frac {2n}{2n-\alpha }}) \in L^{1}({\Omega }, \mu )\) as p=2n/α or the Lebesgue space L∞(Ω,μ) as 2n/α<p≤∞; the geometric characterization of a fractional Sobolev embedding domain via Cα,p(K,Ω). some of the essential behaviours of K with Cα,p(K,Ω)=0.