  A map $\varphi:K\to R^2$ of a graph $K$ is approximable by embeddings, if for
each $\varepsilon>0$ there is an $\varepsilon$-close to $\varphi$ embedding
$f:K\to R^2$. Analogous notions were studied in computer science under the
names of cluster planarity and weak simplicity. This short survey is intended
not only for specialists in the area, but also for mathematicians from other
areas. We present criteria for approximability by embeddings (P. Minc, 1997, M.
Skopenkov, 2003) and their algorithmic corollaries. We introduce the van Kampen
(or Hanani-Tutte) obstruction for approximability by embeddings and discuss its
completeness. We discuss analogous problems of moving graphs in the plane apart
(cf. S. Spiez and H. Torunczyk, 1991) and finding closest embeddings (H.
Edelsbrunner). We present higher dimensional van Kampen obstruction, its
completeness result and algorithmic corollary (D. Repovs and A. Skopenkov,
1998).
