  We consider the question which compact metric spaces can be obtained as a
Lipschitz image of the middle third Cantor set, or more generally, as a
Lipschitz image of a subset of a given compact metric space.
  In the general case we prove that if $A$ and $B$ are compact metric spaces
and the Hausdorff dimension of $A$ is bigger than the upper box dimension of
$B$, then there exist a compact set $A'\subset A$ and a Lipschitz onto map
$f\colon A'\to B$.
  As a corollary we prove that any `natural' dimension in $\mathbb{R}^n$ must
be between the Hausdorff and upper box dimensions.
  We show that if $A$ and $B$ are self-similar sets with the strong separation
condition with equal Hausdorff dimension and $A$ is homogeneous, then $A$ can
be mapped onto $B$ by a Lipschitz map if and only if $A$ and $B$ are
bilipschitz equivalent.
  For given $\alpha>0$ we also give a characterization of those compact metric
spaces that can be obtained as an $\alpha$-H\"older image of a compact subset
of $\mathbb{R}$. The quantity we introduce for this turns out to be closely
related to the upper box dimension.
