Let $F$ be a finitely generated free group, $\bar{F}$ an isomorphic
copy of $F$, $W$ a word in $F$ and $\bar{W}$ its copy in $\bar{F}$.
A Baumslag double is a free product of $F$ and $\bar{F}$ amalgamated
via $\bar{W}=W$. For example, an orientable surface group of genus 2
is a Baumslag double, and it is known that a Baumslag double is a
hyperbolic group if and only if $W$ is not a proper power of in F.

We will discuss what conditions make fundamental groups of orientable
and nonorientable surfaces of finite genus embed into Baumslag
doubles, and also present recent results concerning the Surface Group
Conjecture, which states: Suppose that G is a non-free, non-solvable
one-relator group such that every subgroup of finite index is again a
one-relator group and every subgroup of infinite index is a free
group. Then G is a surface group.