TITLE: What are some of the big open problems in 3-manifold theory?
QUESTION [60 upvotes]: From what I understand, the geometrization theorem and its proof helped to settle a lot of outstanding questions about the geometry and topology of 3-manifolds, but there still seems to be quite a lot of activity.  I am not prepared to make a full push to familiarize myself with the literature in the near future, but I am still curious to know what people are working on, what techniques are being developed, etc.
So I was hoping people could briefly explain some of the main open questions and programs that are motivating current research on 3-manifolds.  I'll let the community decide if this undertaking is too broad, but I'm hoping it is possible to give a rough impression of what is going on.  References to survey articles are appreciated, especially if they are accessible to non-experts like myself.
It seems like the community wiki designation is appropriate for this question, and the usual rules ought apply.

REPLY [7 votes]: Thurston's remaining challenge:
Show that volumes of hyperbolic 3-manifolds are not all rationally related.
THURSTON'S VISION AND THE VIRTUAL FIBERING THEOREM
Is there an algorithm which determines whether two funda-
mental groups of 3-manifolds are isomorphic?
DECISION PROBLEMS FOR 3-MANIFOLDS AND THEIR
FUNDAMENTAL GROUPS