  Our aim is to characterize the homogeneous fractional
Sobolev-Slobodecki\u{\i} spaces $\mathcal{D}^{s,p} (\mathbb{R}^n)$ and their
embeddings, for $s \in (0,1]$ and $p\ge 1$. They are defined as the completion
of the set of smooth and compactly supported test functions with respect to the
Gagliardo-Slobodecki\u{\i} seminorms. For $s\,p < n$ or $s = p = n = 1$ we show
that $\mathcal{D}^{s,p}(\mathbb{R}^n)$ is isomorphic to a suitable function
space, whereas for $s\,p \ge n$ it is isomorphic to a space of equivalence
classes of functions, differing by an additive constant. As one of our main
tools, we present a Morrey-Campanato inequality where the
Gagliardo-Slobodecki\u{\i} seminorm controls from above a suitable Campanato
seminorm.
