A probability measure preserving (p.m.p.) action of a group G is said to be treeable if the orbits of the action can be measurably structured by trees. A countable group G is called treeable if it has a free p.m.p. action which is treeable. The group G is called strongly treeable if all of its free p.m.p. actions are treeable. I will discuss recent joint work with C. Conley, D. Gaboriau, and A. Marks in which we show that finitely generated groups with planar Cayley graphs (e.g., surface groups) are strongly treeable. This provides the first examples of nonamenable strongly treeable groups with one end.