  We study the dynamics of Thurston maps under iteration. These are branched
covering maps $f$ of 2-spheres $S^2$ with a finite set $\mathop{post}(f)$ of
postcritical points. We also assume that the maps are expanding in a suitable
sense. Every expanding Thurston map $f\: S^2 \to S^2$ gives rise to a type of
fractal geometry on the underlying sphere $S^2$. This geometry is represented
by a class of \emph{visual metrics} $\varrho$ that are associated with the map.
Many dynamical properties of the map are encoded in the geometry of the
corresponding {\em visual sphere}, meaning $S^2$ equipped with a visual metric
$\varrho$. For example, we will see that an expanding Thurston map is
topologically conjugate to a rational map if and only if $(S^2, \varrho)$ is
quasisymmetrically equivalent to the Riemann sphere $\widehat{\mathbf{C}}$. We
also obtain existence and uniqueness results for $f$-invariant Jordan curves
$\mathcal{C}\subset S^2$ containing the set $\mathop{post}(f)$. Furthermore, we
obtain several characterizations of Latt\`{e}s maps.
