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: Here a group is called <i> linear</i> if it is isomorphic to a subgroup of $GL(n,\mathbb C)$ for some $n$. 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 <a href="http://arxiv.org/abs/1004.3619">showed</a> 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 Poincaré duality group  is a 3-manifold group?

Comments: This is wide open, but see e.g. <a href="http://front.math.ucdavis.edu/0410.5043">this survey</a> of Wall, and <a href="http://www.maths.usyd.edu.au/u/jonh/pdq.pdf">this list</a> of questions by Hillmann.