  Let $d\ge 1$ be an integer and ${\bf r}=(r_0,\dots,r_{d-1}) \in
\mathbf{R}^d$. The {\em shift radix system} $\tau_\mathbf{r}: \mathbb{Z}^d \to
\mathbb{Z}^d$ is defined by $$ \tau_{{\bf r}}({\bf
z})=(z_1,\dots,z_{d-1},-\lfloor {\bf r} {\bf z}\rfloor)^t \qquad ({\bf
z}=(z_0,\dots,z_{d-1})^t). $$ $\tau_\mathbf{r}$ has the {\em finiteness
property} if each ${\bf z} \in \mathbb{Z}^d$ is eventually mapped to ${\bf 0}$
under iterations of $\tau_\mathbf{r}$. In the present survey we summarize
results on these nearly linear mappings. We discuss how these mappings are
related to well-known numeration systems, to rotations with round-offs, and to
a conjecture on periodic expansions w.r.t.\ Salem numbers. Moreover, we review
the behavior of the orbits of points under iterations of $\tau_\mathbf{r}$ with
special emphasis on ultimately periodic orbits and on the finiteness property.
We also describe a geometric theory related to shift radix systems.
