The geometization conjecture shows that every 3-manifold decomposes into geometric pieces. In some sense, the non-hyperbolic pieces are "well-known" since many decades, whereas the hyperbolic pieces are not. Therefore the current research focuses mainly in "understanding" hyperbolic 3-manifolds. Of course, "understanding" is not a well-defined mathematical problem: however, I think that researchers in the field mostly agree that we are still far from reaching this goal. 

For instance, as opposite to Seifert manifolds, hyperbolic manifolds are not classified in a strict sense: every Seifert manifold has a standard unique "name" which tells many things about its geometry and topology, but hyperbolic manifolds do not have such univoque names. Volumes of hyperbolic manifolds are still poorly understood, and even a simple relationship between manifolds like a "topological covering" is far from being understood, all the conjectures listed by Wilton above show.