Here are two problems on 3-manifold groups (i.e. fundamental groups of compact 3-manifolds) that I find important.
a. Are 3-manifold groups linear?
Comments: A group is called linear if it is isomorphic to a subgroup of $GL(n,\mathbb C)$. One can also ask this over other fields but let's focus on $\mathbb C$. Thurston conjectured that 3-manifold groups are linear, because the geometrization implies that they are residually finite (which is weaker than linearity for finitely generated groups). Aschenbrenner-Friedl recently showed that 3-manifold groups are virtually residually $p$ for all but finitely many $p$'s, which again is known for fg linear groups.
b. Is it true that every 3-dimensional Poincare duality group is a 3-manifold group?
Comments: This is wide open, but see e.g. this survey of Wall, and this list of questions by Hillmann.