  A rational lemniscate is a level set of $|r|$ where $r: \hat{\mathbb{C}}
\rightarrow \hat{\mathbb{C}}$ is rational. We prove that any planar Euler graph
can be approximated, in a strong sense, by a homeomorphic rational lemniscate.
This generalizes Hilbert's lemniscate theorem; he proved that any Jordan curve
can be approximated (in the same strong sense) by a polynomial lemniscate that
is also a Jordan curve. As consequences, we obtain a sharp quantitative version
of the classical Runge's theorem on rational approximation, and we give a new
result on the approximation of planar continua by Julia sets of rational maps.
