  In this article, the authors introduce Besov-type spaces with variable
smoothness and integrability. The authors then establish their
characterizations, respectively, in terms of $\varphi$-transforms in the sense
of Frazier and Jawerth, smooth atoms or Peetre maximal functions, as well as a
Sobolev-type embedding. As an application of their atomic characterization, the
authors obtain a trace theorem of these variable Besov-type spaces.
