There are a number of interesting open problems in continuum theory that hinge on determining which tree-like continua can be embedded in the plane. Up to now, there are very few techniques available to show that a given tree-like continuum cannot be embedded in the plane. However, for a special class of tree-like continua, those which are inverse limits of simplicial inverse systems of trees, there is some hope that an algorithm may exist for checking planarity. I will describe this state of affairs, and pose some questions and computational challenges.

At the same time, recent results are revealing that more tree-like continua can be embedded in the plane than perhaps were expected. I will discuss two such results:

1) Suppose $Y$ is any continuum of the form $Y = X \cup R$, where $X$ is an arc-like continuum, $R$ is a ray, $X \cap R = \emptyset$, and $\overline{R} \setminus R \subseteq X$. Then $Y$ can be embedded in the plane.

2) Suppose $Y$ is any continuum of the form $Y = K \cup \bigcup_{n=1}^\infty A_n$, where $K$ is a Knaster continuum and ${A_n: n = 1,2,\ldots}$ is a family of pairwise disjoint arcs, each intersecting $K$ in a single point, with $\mathrm{diam} A_n \to 0$. Then $Y$ can be embedded in the plane.

This is joint work with Andrea Ammerlaan and Ana Anušić.