A rational lemniscate is a level set of $|r|$ where $r$ is a rational map (a holomorphic self-map of the Riemann sphere). Equivalently, a rational lemniscate is the pull-back of a Euclidean circle under a rational map. Lemniscates "usually" look like collections of pairwise disjoint Jordan curves, but sometimes these curves can intersect, namely when the circle being pulled back contains a critical value of $r$. We prove that any such configuration of curves (intersecting or not) can be approximated, in a suitable sense, to arbitrary precision by a rational lemniscate. This improves on work of Hilbert, and later Walsh-Russell on polynomial lemniscates: their interest was in applications to polynomial approximation and we discuss a related application of our result to rational approximation. This talk is based on joint work with Christopher Bishop and Alexandre Eremenko.