  This paper deals with the fractional Sobolev spaces W^[s,p]. We analyze the
relations among some of their possible definitions and their role in the trace
theory. We prove continuous and compact embeddings, investigating the problem
of the extension domains and other regularity results. Most of the results we
present here are probably well known to the experts, but we believe that our
proofs are original and we do not make use of any interpolation techniques nor
pass through the theory of Besov spaces. We also present some counterexamples
in non-Lipschitz domains.
