  Shifts of finite type and the notion of shadowing, or pseudo-orbit tracing,
are powerful tools in the study of dynamical systems. In this paper we prove
that there is a deep and fundamental relationship between these two concepts.
  Let $X$ be a compact totally disconnected space and $f:X\to X$ a continuous
map. We demonstrate that $f$ has shadowing if and only if the system
$({f},{X})$ is (conjugate to) the inverse limit of a directed system of shifts
of finite type. In particular, this implies that, in the case that $X$ is the
Cantor set, $f$ has shadowing if and only if $(f,X)$ is the inverse limit of a
sequence of shifts of finite type. Moreover, in the general compact metric
case, where $X$ is not necessarily totally disconnected, we prove that $f$ has
shadowing if and only if $({f},{X})$ is a factor of (i.e. semi-conjugate to)
the inverse limit of a sequence of shifts of finite type by a quotient that
almost lifts pseudo-orbits.
