  Let $\sigma$ be a non-unit Pisot substitution and let $\alpha$ be the
associated Pisot number. It is known that one can associate certain fractal
tiles, so-called \emph{Rauzy fractals}, with $\sigma$. In our setting, these
fractals are subsets of a certain open subring of the ad\`ele ring
$\mathbb{A}_{\mathbb{Q}(\alpha)}$. We present several approaches on how to
define Rauzy fractals and discuss the relations between them. In particular, we
consider Rauzy fractals as the natural geometric objects of certain numeration
systems, define them in terms of the one-dimensional realization of $\sigma$
and its dual (in the spirit of Arnoux and Ito), and view them as the dual of
multi-component model sets for particular cut and project schemes. We also
define stepped surfaces suited for non-unit Pisot substitutions. We provide
basic topological and geometric properties of Rauzy fractals associated with
non-unit Pisot substitutions, prove some tiling results for them, and provide
relations to subshifts defined in terms of the periodic points of $\sigma$, to
adic transformations, and a domain exchange. We illustrate our results by
examples on two and three letter substitutions.
